HER  BOOK 


i  * 


DABQLL'8 

SCHOOLMASTER'S    ASSISTANT. 

IMPROVED  AND.  ENLAK9*D. 

' 

^V'    BEIWG  A 
*>J^ 

TACTICAL  SYSTEM 

OF 

ARITHMETICS. 

ADAPTED  TO 

THE  UNITED  STATES. 


BY   NATHAN   DABOLL. 


WITH  THE  ADDITION  OF  THE 

FARMERS    AND  ME^HANICKS'  BEST 

IETHOD    OF    B00M=KEBP 

DESIGNED  AS   A 

COMPANION  TO  DABOLL'S  ARITHMETICK. 
BY  SAMUEL  GREEN. 


MIBDLETOWN,  (Con.) 

PUBLISHED  BY  WILLIAM    H.  NILE.S. 
Stereotyped  by  A.  Chandler,  New- York. 


District  of  Cennecticutj  ss. 
BE  IT  R&MEMBKRED,  That  ton  Hie  tw*my-eij;Uih  day  of  September,  in  the  lor  ty- 

' 


.,     ~      fifth  year  of  the  IpdtpeuikuM  ol'  ih*  -U±ii4*tf  SktU*  of  AntMriea,  Samuel  Gr««li, 
'•'  of  »aitl  District,  hath  deposited  ia  ^iia  offie*  the  utl»  of  a  Book,  th»  right 


\vneieofhe  ctilms  a»  yropri«tor}  in  tfie  wocds  fiDUowiiig,  to  wU  :— 

((  Daftoll's  Sclipoltnastec's  Assistant,  improved  and  cularged.  Being  a  plain  prac- 
tical system  of  Arithmetick.  Adapted  to  the  United  States.  By  Nathan  Daboll." 

In  conformity  to  the  Act  of  the  Congress  of  the  United  States,  entitled,  "  An  Act 
for  the  encouragement  of  learning,  by  securing  the  Copies  of  Maps,  Charts,  and  Books, 
to  the  authors  and  proprietors  of  then),  during  the  times  therein  mentioned." 

HENRY  W.  EDWARDS, 
Clerk  of  the  District  of  Connecticut. 
A  true  copy  ef  Record  :  Examined  and  sealed  by  me, 

H.  W.  EDWARDS, 
Cftrk  of  the  District  of  Connecticut. 


PSVCH. 

I.JRRARV 


GIF? 


RECOMMENDATIONS, 

EDUC.- 


Yak  GoUege,  &»».  27,  1799. 

I  HAVE  read  DABOLL'S  SCHOOLMASTER'S  ASSISTANT. 
The  arrangement  of  the  different  branches  of  Arithmetic 
is  judicious  and  perspicuous.  The  author  has  well  ex- 
plained Decimal  Arithmetic,  and  has  applied  it  in  a  plain 
and  elegant  manner  in  the  solution  of  various  questions, 
and  especially  to  those  relative  to  the  Federal  Computation 
of  money.  I  think  it  will  be  a  very  useful  book  to  School- 
masters and  their  pupils. 

JOSIAH  MEIGS,  Professor  of  Mathematics 

and  Natural  Philosophy. 
[Now  Surveyor-General  of  the  United  States.,] 


I  HAVE  given  some  attention  to  the  work  above  men- 
tioned, and  concur  with  Mr.  Professor  Meigs  in  his  opinion 
of  its  merit.  NOAH  WEBSTER. 

New-Haven  Dec.  12,  1799. 


Rhode-Island  College,  Nov.  39,  J799. 

1  HAVE  run  through  Mr.  DABOLL'S  SCHOOLMASTER'S 
ASSISTANT,  and  have  formed  of  it  a  very  favourable  opinion. 
According  to  its  original  design,  I  think  it  well  "  calculated 
to  furnish  Schools  in  general  with  a  methodical,  easy,  and 
comprehensive  System  of  Practical  Arithmetic."  I  there- 
fore hope  it  may  find  a  generous  patronage,  and  have  an 
r xtensive  spread. 

ASA  MESSER,  Professor  of  the  Learned  Languages, 

and  teztchcr  of  Dfathc/natirs. 
r\ow  PreSirt*rft  of  that  Tn^itutian.«! 

053 


ft  r-:  c  o  -  j  M  K  x  D  A  T  r  o  x  s . 

40  Piainfield  Academy,  April  20,  1802. 

l<  MARK  use  of  DABOLT/S  SCHOOLMASTER'S  ASSISTANT, 
ia  teaching  common  Arithmetic,  and  think  it  the  best  cal- 
culated for  that  purpose  of  any  which  has  fallen  within  mv 
observation.  JOHN  ADAMS, 

Rector  of  Plain  field  Academy. 

[New  Principal  of  Philips'  Academy,  Andover,  Mass.] 


ELilkrlca  Academy,  (Mass.)  Dec.  10,  1807. 

HAVING  examined  Mr.  DABOLL'S  System  of  Arithmetic, 
]'  am  pleased  with  the  judgment  displayed  in  his  method, 
and  the  perspicuity  of  his  explanations,  and  thinking  it  as 
easy  and  comprehensive  a  system  as  any  with  which  I  am 
acquainted,  can  cheerfully  recommend  it  to  the  patronage 
of  rnstnirtw.  SAMUEL  WHITING, 

Teacher  of  Mathematics. 


Prom  Mr.  Kennedy,  Teacher  of  Mathematics. 

I  BECAME  acquainted  with  DABOLL'S  SCHOOLMASTER'S 
ASSISTANT,  in  the  year  1802,  and  on  examining  it  atten- 
tively, gave  it  my  decisive  preference  to  any  other  system 
extant,  and  immediately  adopted  it  for  the  pupils  under  my 
charge  ;  and  since  that  time  have  used  it  exclusively  in 
elementary  tuition,  to  the  great  advantage  and  improve- 
ment of  the  student,  as  well  as  the  ease  and  assistance  of 
the  preceptor.  I  also  deem  it  equally  well  calculated  for 
the  benefit  of  individuals  in  private  instruction  ;  and  think 
it  my  duty  to  give  the  labour  and  ingenuity  of  the  author 
the  tribute  of  my  hearty  approval  and  recommendation. 

ROGER  KENNEDY. 

New- York,  March  90, 


PHM. 


THE  design  of  this  work  is  to  furnish  rh;.'  ->f  UH; 

United  States  with  a  methodical  and  comprehensive  .v, 
of  Practical  Arithmetic,  in  which  1    have  endeavoured, 
through  the  whole,  to  have  the  rales  as  concise  and  fami- 
liar as  the  nature  of  the  .subject  will  permit. 

During  the  long  period  which  I  have  devoted  to  tho  in- 
struction of  youth  in  Arithmetic,  I  have  made  use  of vanou ; 
systems  which  have  just  claims  to  scientific  merit;  but  \\i\>. 
authors  appear  co  have  been  .deficient  in  an  important 
point — the  practical  teacher's  experience.  They  have  b 
too  sparing  of  examples,  especially  in  the  first  rudiments; 
in  consequence  of  which,  the  young  pupil  is  hurried  through 
the  ground  rules  too  fast  for  his  capacity.  This  objection 
I  have  endeavoured  to  obviate  in  the  fallowing  tronti.se. 

In  teaching  the  first  rules,  1  have  found  it  best  to  en- 
courage the  attention  of  scholars  by  a  variety  of  easy  and 
familiar  questions,  which  might  serve  to  strengthen  their 
minds  as  their  studies  grew  more  arduous. 

The  rules  arc  arranged  ia  such  order  as  to  in!. 
most  simple  and  necessary  parts,  previous  to  those  which 

are  more  abstruse  and  difficult. 

To  enter  into  a  detail  of  the  whole  work  would   • 

tlious  ;  I  shall  therefore  notice  only  a  fo\v  particulars, 

refer  the  reader  to  the  contents. 

Although  the  Federal   Coin  is  purely  decimal,  it 

nearly  allied  to  whole  numbers,  and  so  absolutely  u 

to  be  understood  by  every  one,  that  I  have  intro 

immediately  after   addition  of  whole    nurnboivi,  tiud 

shown  how  to  find  the  value  of  goods  therein,  im?;; 

after  simple  multiplication  ;  which  niny  b 

tage  to  many,  who  perhaps  will  not  1\ 

learning  fractions* 

In  the  arrangement  of  fraction 

new  mwhod,  the    advantages  and   facility   of  which 

sufficiently  npotoo-i/o  for  its  not  her: 


PREFACE. 

systems.  As  decimal  fractions  may  be  learned  much  easier 
than  vulgar,  and  are  more  simple,  useful,  and  necessary, 
rind  soonest  wanted  in  more  useful  branches  of  Arithmetic, 
they  ought  to  be  learned  first,  and  Vulgar  Fractions  omitted, 
until  further  progress  in  the  science  shall  make  them  ne- 
cessary. It  may  be  well  to  obtain  a  general  idea  of  them, 
and  to  attend  to  two  or  three  easy  problems  therein  ;  after 
which,  the  scholar  may  learn  decimals,  which  will  be  ne- 
ressary  in  the  reduction  of  currencies,  computing  interest, 
and  many  other  branches. 

Besides,  to  obtain  a  thorough  knowledge  of  Vulgar  Frac- 
tions, is  generally  a  task  too  hard  for  young  scholars  who 
have  made  no  further  progress  in  Arithmetic  than  Reduc- 
lion,  and  often  discourages  them. 

I  have  therefore  placed  a  few  problems  in  Fractions,  ac- 
cording to  the  method  above  hinted ;  and  after  going  through 
the  principal  mercantile  rules,  have  treated  upon  Vulgar 
Fractions  at  large,  the  scholar  being  now  capable  of  going 
through  them  with  advantage  and  ease. 

In  Simple  Interest,  in  Federal  Money,  I  have  given  seve- 
ral new  and  concise  rules  ;  some  of  which  are  particularly 
designed  for  the  use  of  the  compting-house. 

The  Appendix  contains  a  variety  of  rules  for  casting 
Interest,  Rebate,  &-c.  together  with  a  number  of  the  most 
easy  and  useful  problems,  for  measuring  superficies  and 
solids,  examples  of  forms  commonly  used  in  transacting 
business,  useful  tables,  &c.  which  are  designed  as  aids  in 
the  common  business  of  life. 

Perfect  accuracy,  in  a  work  of  this  nature,  can  hardly 
be  expected  ;  errors  of  the  press,  or  perhaps  of  the  author, 
may  have  escaped  correction.  If  any  such  are  pointed 
out,  it  will  be  considered  as  a  mark  of  friendship  and  fa- 
veur,  by 

The  public's  most  humble 

and  obedient  Servant, 

NATHAN  DABOLL. 


TABLE  OF  CONTENTS. 


ADDITION,  Simple,      -     •    -         -         -        -         -         -•- 

-  of  Federal  Moray,      ..... 

-  Compound,        ------- 

Alligation, 

Annuities  or  Pensions,  at  Compound  Interest,      - 
Arithmetical  Progression,  ------ 

Barter,       -        -        -         -        - 

Brokerage,          -                                               ...         -  113 

Characters,  Explanation  of,          ------  14 

Commission,      -          ------         --1  12 

Conjoined  Proportion,         ------- 

Coins  of  the  United  States,  Weights  of,       -         -         -      '-  220 

Division  of  Whole  Numbers,  32 

—  -  --  Contractions  in,  36 

-  Compound,  S3 
Discount,            -         ........  123 

Duodecimals,              --------  216 

Ensurance,         ---------H4 

Equation  of  Payments,       -------  126 

Evolution  or  Extraction  of  Roots,       -         -         -         -•       -  167 

Exchange,          _--_--_-.  139 

Federal  Money,  21 

--  Subtraction  of,            -         -         -         -         .  55 

Fellowship,         ------.-_  \<%fc 

-  Compound,      -------  134 

Fractions,  Vulgar  and  Decimal,          -  69,  143 

Interest,  Simple,                  ---....  593 

by  Decimals,     ------  157 


Compound, 

by  Decimals, 
* 


Inverse  Proportion*, 

Involution,         ---------  jgg 

Lose  and  Gain,          -----_.. 

Multiplication,  Simple,       -------  yj 

-  -  Application  and  Use  of,        -        ...  30 

-  Supplement  to,                                -        .        _  37 

-  —  Compound,          -                  -        -        -        .  43 
Numeration,       ---•---.. 

I™?*06*             .........  99 

Position,             ---------  rgg 

Permutation  of  Quantities,          -         -         -         -         -         ..  195 


VlU  i  \iJLK  OF   i  0\  TK.N  i'.-; . 

Questions  for  Exorcise,       ---.__ 

Reduction,          -_._....._.  59 

of  Currencies,  do.  of  Coin,          -         -         -         -82,86 

Rule  of  Three  Direct,  do.  Inverse,      -                                     -  90,97 

Double,        -         -         -         -         -         -         -  136 

Rules  for  reducing  the  different  currencies  of  the  several  United 
States,  also  Canada  and  Nova  Scotia,  each  to  the  par 

of  all  others,  8$ 

Application  of  the  preceding,     -----  89 

:-  Short  Practical,  for  calculating  Interest,     -  114 

•  for  casting  Interest  at  6  per  cent,        -  203 

for  finding  the  contents  of  Superfices  and  Solids,         -  208 

to  reduce  the  currencies  of  the  different  States  to  Fede- 
ral Money,       --------  200 

Rebate,  a  short  method  of  finding  the,  of  any  given  .sum,  for 

months  and  days,     ----__*  205 

Subtraction,  Simple,  23 

— Compound,     -  43 

Table,  Numeration  and  Pence,             -  9 

Addition,  Subtraction,  and  Multiplication,          -  10 

of  Weight  and  Measure,            -  if 

of  Time  and  Motion,         -         -         -         -               "     •»  13 

showing  th-o  number  of  days  from  any  day  of  one  month 

to  the  same  doy  in  any  other  month,          -  160 

r-  showing  the  amount  of  I/,  or  1  dollar,  at  5  and  6  per 

cent.  Compound  Interest,  for  20  years,      -  220 

showing  the  amount  of  I/,  annuity,  forborne  for  31  years 

or  under,  at  5  and  6  per  cent.  Compound  Interest,      -         221 

showing  the  present  worth  of  II.  annuity,  for  31  years, 

at  5  and  6  per  cent.  Compound  Interest,     -  221 

i  of  Cents,  answering;  to  the  currencies  of  the  United 

States,  with  Sterling,  fcc.          -----         224 
.      showing  the  value  of  Federal  Money  in  other  currencies,       225 

Tare  and  Tret,  -         - 10S 

Useful  Forme  in  transacting  business,  -  226 

Weights  of  several  pieces  of  English,  Portuguese,  and  French 

Gold  Coins,  in  dollars,  cents,  and  mills,    - 

of  English  and  PortuffUftse  Gold,         do.        de.     -        f«S 

^ — —  of  French  and  Spanish  Gold,  do.        do. 


DABOLL'S 


SCHOOLMASTER'S  ASSISTANT. 


ARITHMETICAL  TABLES. 


Numeration  Table. 

Pence  Table. 

en 

d.     s.  d. 

d. 

5. 

j3 

20isl     8 

12 

is  1 

C/3 

03 

30    2     6 

24 

.2 

.    . 

O 

1 

40    3     4 

36 

3 

i 

c 
o 

EH 

I 

50    4    2 

48 

4 

.  >H 

0 

60    5    0 

60 

5 

0 

QB 

§ 

o 

EH 

en 

CO 

70     5  10 

72 

6 

«*-! 

o 

QQ 

c 

J 

o 

& 

ctf 

T3 
2 

80     6     8 

84 

7 

"O 

01 

o 

co 

3 

QQ 

03 

90     7     6 

96 

8 

a 

a 

9 

1 

8 
9 

3 
g 
7 
8 

c 

a 

6 

7 

a 

£ 

5 
6 

O 

EH 
4 
5 

3 

a 

3 
4 

fl 

0 

H 
3 

1 

2 

100     8     4 
110     9    2 
120  10     0 

108 
120 
132 

9 
10 

U 

9 

8 

7 

8- 

6 

5 

4 

3 

9 

7 
8 

7 

5 

6 

4 
5 

9 

8 

7 

6 

make. 

9 

Q 

7 

9 

8 
9 

4  farthings  1  penny 
12  pence  1  shilling, 
20  shilling.  1  pound. 

£.f! 

10 


ARITHMETICAL  TABLES. 


ADDITION  AND  SUBTRACTION  TABLE. 


1 

* 

3 

4 

5  |  6  |  7|  8  |  9  |  10 

11 

12 

2 

4 

5 

6 

7  |  8  |  9  |  10  |  11  !  12 

13 

14 

3 

5 

6 

7 

8  |  9  |  10  |  11  |  12  |  13 

14 

15 

4 

6 

7 

8 

9  |  10  |  11  |  12  |  13  |  14 

15 

16 

5 

7 

8 

9 

10  |  11  |  12  |  13  |  14  |  15 

16 

17 

6 

8 

9 

10 

11  |  12  |  13  |  14  |,15  |  16 

17 

18 

7 

9 

10 

11 

12  |  13  |  14  |  15  |  16  |  17 

18 

19 

8 

10 

11 

12 

13  |  14  |  15  |  16  |  17  |  18 

19 

20 

9 

11 

12 

13 

14  |  15  |  16  |  17  |  18  |  19 

20 

21 

10 

12 

13 

14 

15  |  16  |  17  |  18  |  19  |  20 

|21 

|22 

MULTIPLICATION  TABLE. 


I 

"2~ 

2 
4 

3|  4 

5|  6 

7 

8 

9 

10 

11 

12 

6|  8 

10  |  12 

14 

16 

18 

20 

22 

24 

3 

6 

9  |  12 

15  |  18 

21 

24 

27 

30 

33 

36 

4 

IT 

8 

12  |  16 

20  |  24 

28 

32 

36 

40 

44 

48 

'60 

10 

15  |  20 

25  |  30 

35 

40 

45 

50  |  55 

6 

12 

18|  24 

30  |36 

42 

48 

54 

60 

<*) 

7* 

7 

14 

21  |  28 

35  I  42 

49 

56 

.63 
-75- 

70 

77 

84 

8 

I"5" 

16 

T8- 

24  I  32 

40  |  48 

5(5 

64 

80 

88 

96 

27|36 

45  |  54 

63 

72 

81 

90 

99 

108 

10 

20 

30  |  40 

50  (60 

70 

bO 

90 

100 

111) 

1201 

11 

22 

33|44 

55  |  66 

yy 

88 

^ 

110  |  121 

132 

112 

24 

36|48 

60  |  72 

84 

96 

108 

120  |  132 

144 

To  learn  this  Table  :  Find  your  multiplier  in  the  left 
hand  column,  and  the  multiplicand  a-top,  and  in  the  com- 
mon angle  of  meeting,  or  against  your  multiplier,  along  at 
the  right  hand,  and  under  your  multiplicand,  you  will  find 
the  product,  or  answer. 


ARITHMETICAL  TABLES. 

2.    Troy  Weight. 

24  grains  (gr.)  make       1   penny-weight,  marked        pwt+ 
20  penny- weights,  1  ounce,  oz. 

12  ounces,  1  pound,  lb» 

3.  Avoirdupois  Weight. 

16  drams  (dr.)  make                 1  ounce,  oz. 

16  ounces,                                  1  pound,  //;. 

28  pounds,  1  quarter  of  a  hundred  weight,  qr. 

4  quarters,                               1  hundred  weight,  cwt. 

20  hundred  weight,                   1  tun.  T. 

By  this  weight  are  weighed  all  coarse  and  drossy  goods, 

grocery  wares,  and  all  metals  except  gold  and  silver. 

4.  Apothecaries  Weight. 

20  grains  (gr.)  make  1  scruple,  B 

3  scruples,  1  dram, 
8  drams,                              1  ounce, 

12  ounces,  1  pound, 

Apothecaries  use  this  weight  in  compounding  their  me- 
dicines. 

5.  Cloth  Measure. 

4  nails  (na.)  make  1  quaiter  of  a  yard,  qr. 

4  quarters,  1  yard,  yd. 

3  quarters,  1  Eli  Flemish,  E.  FL 

5  quarters ,  1  Ell  English,  E.  E. 

6  quarter,  1  Ell  French,  E.  Fr, 

6.  Dry  Measure. 

2  pints,  (pt.)  make  1  quart,  qt. 

8  quarts,  1  p  ,ck,  pk. 

4  pecks,                               1  hushel,  bu. 
This  measure  is  applied  to  grain,  beans,  flax-seed,  salt, 

oats,  oysters,  coal,  fyc. 


ARITHMETICAL  TABLES. 


Wine  Measure. 


4  gills  (gi.)  make 

2  pints, 

4  quarts, 
3l£  gallons, 
42  gallons, 
63  gallons, 

2  hogsheads, 

2  pipes, 


pint, 

quart, 

gallon, 

barrel, 

tierce, 

hogshead, 

pipe, 

tun, 


pt. 

qt 

gal. 

bl 

tier. 

Jihd 

f. 


All  brandies,  spirits,  mead,  vinegar,  oil,  &c.  are  measur- 
ed by  wine  measure.  Note.  231  solid  inches,  make  a  gal- 
lon. 


8.  Long  Measure. 


3  barley  corns  (b.  c.)  make     1 

12  incites,  1 

3  feet,  1 

5J  yards, 
40  rods, 

8  furlongs,  1 

3  miles,  1 

69  J  statute  miles,  1 


inch,  marked 
foot, 
yard, 

1  rod,  pole,  or  perch, 
furlong, 
mile, 
league, 
degree,  on  the  earth. 


ifi. 

fl. 
yd. 
rd. 
fur. 
in. 
lea. 


360  degree?,  the  circumference  of  the  earth. 

The  use  of  long  measure  is  to  measure  the  distance  of 
places,  or  any  other  thing,  where  length  is  considered,  with- 
out regard  to  breadth. 

N.  B.  In  measuring  the  height  of  horses,  4  inches  make 
1  hand.  In  measuring  depths,  6  feet  make  1  fathom  or 
French  toise.  Distances  are  measured  by  a  chain,  four 
rods  long,  containing  one  hundred  links. 


ARITHMETICAL  TABLES.  13 

9.  Land,  or  Square  Measure. 

144  square  inches  make  1  square  foot. 

9  square  feet,  1  square  yard. 

30£  square  yards,  or  )  ^ 

o^ot  i    <  *  square  rod. 

272£  square  feet,          f 

40  square  rods,  1   square  rood. 

4  square  roods,  1  square  acre, 

640  square  acres,  1  square  mile. 

10.  Solid,  or  Cubic  Measure. 

1728  solid  inches  make  1  solid  foot. 

40  feet  of  round  timber,  or  > 

Kt\     C       4        f    U  '        U  1     tU11    O1*   lOad. 

50  feet  of  hewn  timber,       ) 
128  solid  feet  or  8  feet  long,  .  l      rf   f        d 

4  wide,  and  4  high,  J 

All  solids,  or  things  that  have  length,  breadth,  and  depth, 
are  measured  by  this  measure.  N.  B.  The  wine  gallon 
contains  231  solid  or  cubic  inches,  and  the  beer  gallon,  282. 
A  bushel  contains  2150,42  solid  inches. 

11.    Time. 

60  seconds  (S.)  make  1  minute,  marked       M. 

60  minutes,  1  hour,  k. 

24  hours,  1  day,  d. 

7  days,  I  week,  «?/ 

4  weeks,  I  month,  mo. 

13  months,  1  day  and  6  hours,  1  Julian  year,  yr. 

Thirty  days  hath  September,  April,  June,  and  November, 
February  twenty-eight  alone,  all  the  rest  have  thirty-one. 
N.  B.  In  Bissextile,  or  leap  year,  February  hath  29  days. 

12.    Circular  Motion. 

60  seconds  (")  make  1  minute, 

60  minutes,  I  degree, 

30  degrees,     .  I  sign,  #. 

12  signs,  or  360  degrees,  the  whole  great  circle  of  the 
7odiack. 


14  CHARACTERS. 

Explanation  of  Characters  used  in  this  Book. 


=  Equal  to,  as  12df.  =  Is.  signifies  that  12  pence  are  equal 
to  1  shilling. 

+  More,  the  sign  of  Addition;  as,  5+7=12,  signifies  that 
5  and  7  added  together,  are  equal  to  12. 

—  Minus,  or  less,  the  sign  of  Subtraction ;  as,  6 — 2=4,  sig- 
nifies that  2  subtracted  from  6,  leaves  4. 

X  Multiply,  or  with,  the  sign  of  Multiplication ;  as, 
4  X  3=12,  signifies  that  4  multiplied  by  3,  is  equal  to  12. 

-'-  The  sign  of  Division  ;  as,  8H-2=4,  signifies  that  8  di- 
vided by  2,  is  equal  to  4;  or  thus,  f =4,  each  of  which 
signify  the  same  thing. 

:  :  Four  points  set  in  the  middle  of  four  numbers,  denote 
them  to  be  proportional  to  one  another,  by  the  rule  of 
three  ;  as  2  :  4  :  :  8  :  16 ;  that  is,  as  2  to  4,  so  is  8  to  16. 

^  Prefixed  to  any  number,  supposes  that  the  square  root  of 
that  number  is  required. 

V  Prefixed  to  any  number,  supposes  the  cube  root  of  that 
number  is  required. 

V  Denotes  the  biquadrate  root,  or  fourth  power,  <fcc. 


ARITHMETIC 


ARITHMETIC  is  the  art  of  computing  by  numbers, 
and  has  five  principal  rules  for  its  operation,  viz.  Numera- 
tion, Addition,  Subtraction,  Multiplication,  and  Division. 


NUMERATION. 

Numeration  is  the  art  of  numbering.  It  teaches  to  ex- 
press the  value  of  any  proposed  number  by  the  following 
characters,  or  figures : 

1,  2,  3,  4,  5,  6,  7,  8,  9,  0— or  cipher. 

Besides  the  simple  value  of  figures,  each  has  a  local 
value,  which  depends  upon  the  place  it  stands  in,  viz.  any 
figure  in  the  place  of  units,  represents  only  its  simple  value, 
or  so  many  ones ;  but  in  the  second  place,  or  place  of  tens,  it 
becomes  so  many  tens,  or  ten  times  its  simple  value  ;  and  in 
the  third  place,  or  place  of  hundreds,  it  becomes  a  hundred 
times  its  simple  value,  and  so  on,  as  in  tfie  following 

Jfote. — Although  a  cipher  standing  alone  signifies  nothing  ;  yet  when  it 
is  placed  on  the  right  hand  of  figures,  it  increases  their  value  in  a  tenfold 
proportion,  by  throwing  them  into  higher  places.  Thus,  2  with  a  cipher  an- 
nexed to  it,  becomes  20,  twenty,  and  with  two  ciphers,  thus,  200, two  hundred. 

2.  When  numbers  consisting  of  many  figures,  are  given  to  be  read,  it 
will  be  found  convenient  to  divide  them  into  as  many  periods  as  we  can,  of 
six  figures  each,  reckoning  from  the  right  hand  towards  the  left,  calling  the 
first  the  period  of  units,  the  second  that  of  millions,  the  third  billions,  the 
fourth  trillions,  &c.  as  in  the  following  number  : 

8073625462789012506792 


4.    Period  of 
Trillions. 


8073 


Period  of 
Billions. 


625462 


Period  of      II.    Period  of 

„«•.,..  "  -,      ..        «/ 


Millions. 
789012 


Units. 
506792 


The  foregoing  number  is  read  thus — Eight  thousand  and  seventy-three 
trillions  ;  six  hundred  and  twenty-five  thousand,  four  hundred  and.  sixty- 
two  billions ;  seven  hundred  and  eighty-nine  thousand  and  twelve  millions  ; 
five  hundred  and  six  thousand  seven  hundred  arid  ninety-two. 

N.  B.     Billions  is  substituted  for  millions  of  millions. 

Trillions  for  millions  of  millions  of  millions. 

Quatrillions  for  millions  of  millions  of  millions  of  millions,  fyc< 


NUMERATION. 

TABLE. 


BMtfEMll,  - 

std       o  o  g--M    ,     , 

!|~     •    '    l-One 
I*  fi,  i    '21  -Twenty-one. 
~   -^   i    321  -Three  hundred  twenty-one. 
'•4321  -Four  thousand  321. 
1    •    •    '54321  -Fifty-four  thousand  321. 
•••654321  -654  thousand  321. 
1    '    7654321-7  million  654  thousand  321. 
•    87654321  -87  million  654  thousand  32U 
987654321  -987  million  654  thousand  321. 
1  2  3  4  5  6  7  8  9  -123  million  456  thousand  789. 
987654348  -987  million  654  thousand  348. 

To  know  the  value  of  any  number  of  figures  : 
RULE. — 1.  Numerate  from  the  right  to  the  left  hand,  each  figure  in 
its  proper  place,  by  saying,  units,  tens,  hundreds,  £c.  as  in  the  Nume- 
ration Table. 

2.  To  the  simple  value  of  each  figure,  join  the  name  of  its  place, 
beginning  at  the  left  hand,  and  reading  to  the  right. 
EXAMPLES. 

Read  the  following  numbers. 
365,  Three  hundred  and  sixty-five. 
5461,  Five  thousand  four  hundred  and  sixty-one. 
1234,  One  thousand  two  hundred  and  thirty-four. 
54026,  Fifty-four  thousand  and  twenty-six. 
123461,  One  hundred  and  twenty-three  thousand   four 

hundred  and  sixty-one. 

4666240,  Four  millions,  six  hundred  and   sixty-six  thou- 
sand two  hundred  and  forty. 

NOTE.     For  convenience  in  reading  large  numbers,  they 
may  be  divided  into  periods  of  three  figures  each,  as  follows  : 

987,  Nine  hundred  and  eighty- seven. 
987  000,  Nine  hundred  and  eighty-seven  thousand. 
987  000  000,  Nine  hundred  and  eighty-seven  million. 
987  654  321,  Nine  hundred  and  eighty-seven  million,  six 
hundred  and  fifty-four  thousand,  three  hun- 
ilred  qnd  twentv-one, 


i,K  ADDITION,  1  t 

To  write  numbers. 

KULE. — Begin  on  the  right  iiand,  write  units  in  the  units  place 
tens  in  the  tens  place,  hundreds  in  the  hundreds  place,  and  so  on 
towards  the  left  hand,  writing  each  figure  according  to  its  proper  value 
in  numeration  ;  taking  care  to  supply  those  places  of  the  natural 
order  with  ciphers  which  are  omitted  in  the  question. 

EXAMPLES. 

Write  down  in  proper  figures  the  following  numbers  : 

Thirty-six. 

Two  hundred  and  seventy-nine. 

Thirty-seven  thousand,  five  hundred  and  fourteen. 

Nine  millions,  seventy-two  thousand  and  two  hundred* 

Eight  hundred  millions,  forty- four  thousand  and  fifty-five. 

SIMPLE  ADDITION. 

IS  putting  together  several  smaller  numbers,  of  the  same 
denomination,  into  one  larger,  equal  to  the  whole  or  sum 
total ;  as  4  dollars  and  6  dollars  in  one  sum  is  10  dollars. 

RULE. — Having  placed  unite  under,un<ls,tens  under  tens,  &c.  draw 
a  line  underneath,  and  begin  with  the  units  ;  after  adding  up  every 
figure  in  that  column,  consider  how  many  tens  are  contained  in  thefr 
sum  ;  set  down  the  remainder  under  the  units,  and  carry  so  many  as 
you  have  tens,  to  the  next  column  of  tens  ;  proceed  in  the  same  man- 
ner through  every  column  or  row,  and  set  down  the  whole  amount 
of  the  last  row. 

EXAMPLES. 

(1.)  (2.)  (3.)  (4.) 

W2      CO 

.  TO                                    3    S3    ^      . 

tn  *"O    vi                              o    o  "O    *» 

T3  C  T5                             ,«£  -C    C  ""O 

<D  65    ®                                        ,    eg    CP 

.      .                i  .      .                 <»  ^'  .      .                        Ht«i.« 

§  •«  S    1  -33  2    £    I  "I.  *0  *0  J    §    g  J 


42  414  1756  552621 

53  291  0432  346977 

52  851  9478  413339 

13  152  1666  321012 

89  698  7422  876543 


+- 


SIMPLE  ADDITION, 


(5.) 

(6.)         (7.) 

3 

1  4 

8 

5 

6 

4 

7 

9 

37145 

6 

7  2 

3 

7 

2 

5 

7 

1 

2 

5  1714 

4 

2  7 

1 

9 

8 

4 

1 

9 

4 

60845 

9 

7  1 

4 

5 

3 

2 

5 

1 

6 

3  ?'  857 

3 

2  8 

5 

1 

7 

1 

4 

3 

2 

61784 

1 

4  5 

7 

2 

3 

2 

7 

1 

9 

52101 

6  4 

7 

2*7  3 

8 

(9.) 
4  1  2 

8 

(10.) 
5263 

1  7 

8  4 

5 

9 

3 

7 

1 

4 

2719 

6 

3  7 

256 

3 

7 

1 

4 

7 

3841 

9 

2  5 

4  1 

7 

1 

8 

3 

2 

1 

5319 

2 

6  1 

7  2 

3 

7 

1 

4 

3 

7 

6108 

4 

3  8 

4  1 

9 

5 

1 

7 

2 

6 

3719 

5 

7  2 

8  4 

3 

7 

2 

5 

1 

3 

2914 

7 

(11.) 

— 

- 

(12.) 

9 

4  2 

3 

1'7  8 

2 

9 

3  7 

1845687 

7 

4  2 

1 

061 

0 

8 

5  1 

1704229 

6 

1  0 

0 

4  2  7 

9 

6 

I 

9466372 

7 

6  2 

3 

1  4  5 

7 

2 

8340734 

2 

0  0 

0 

4  1  2 

3 

4 

270155 

7 

0  4 

1 

360 

5 

3 

36023 

5 

6  7 

8 

093 

8 

7 

1950 

(13.) 

(14.) 

9  6 

2  4 

3  0 

6 

4  6 

2590 

0 

4 

6  2 

8  1 

4 

5  1 

34004 

5 

2  1 

6  0 

4 

3  2 

540443 

3 

8 

7  6 

1  0 

4 

2  5 

3705532 

6 

3 

4  6 

2 

1  4 

405217 

4 

4  0 

3 

0  9 

40647626 

9 

9 

8 

2  7 

206859 

Ir 

IT  To  prove  Addition,  begin  at  the  top  of  the  sum,  and   reckon 
the  figures  downwards  in  the  same  manner  as  they  were  added  up- 


LE  ADDITION.  19 

wards,  and  if  it  be  right,  this  sum  total  will  be  equal  to  the  first :  Or 
cut  off  the  upper  line  of  figures,  and  find  the  amount  of  the  rust;  then 
if  the  amount  and  upper  line,  when  added,  be  equal  to  the  total,  the 
work  is  supposed  to  be  right. 

2.  There  is  another  method  of  proof,  as  follows  : — 

Reject  or  oast  out  the  nines  in  each  row     EXAMPLE. 
or  sum  of  figures,  and   set  down  the   re-     3  7  8  2  |  ^w  2 
maiuders,  each  directly  even  with  the  figures     5  7  6  0  \  &  6 


in  its  row  ;  find  the  sum  of  these  remain-     8755 


%  7 


ders  ;  then    if  the   excess  of  nines  in  the     

sum  found  as  before,  is  equal  to  the  excess  18  3  0  3  i   S  6 

of  nines  in  the  sum  total,  the  work  is  sup-     j  £j  -. 

posed  to  be  right. 

15.  Add  8635,  2194,  7421,   5063,  2196,  and    1245,  to- 
gether. Ans.  26754. 

16.  Find   the  sum  of  3482,   783645,   318,   7530,  and 
9678045.  Ans.   10473020. 

17.  Find  the  sum  total  of  604,  4680,  9ri,  64,  and  54, 

live  hundred. 

18.  What  is  the  sum  total  of  24674, 16742,  34678,  10467, 
and  134-39?  .       Ans..  One  hundred  thousand. 

19.  Add  1021,  3439,  28703,  289,  and  6438,  together. 

Ans.  Forty  thousand. 

20.  What  is  the  sum  total  of  the  following  numbers,  viz. 
2340,  1006,  3700,  and  4005  ?  Ans.  11111. 

21.  What  is  th«  sum  total  of  the  following  numbers,  viz. 
Nine  hundred  and  forty-seven, 

Seven  thousand  s:  v  hundred  and  five, 
Forty-five  thousand  six  hundred, 
Three  hundred  and  eleven  thousand, 
Nine  millions,  u:id  twenty-five, 
Fifty-two  millions,  and  nine  thousand  ? 


Answer,  61374177 

Required  the  sum  of  the  following  numbers,  viz. 
Five  hundred  and  sixty-eight, 
Eight  thousand  eight  hundred  and  five, 
Seventy-nine  thousand  si?:  hundred, 


FKDKHAL  M< 


Nine  hundred  and  eleven  thousand, 
Nine  millions  and  twenty-six. 


Answer,  9999999 


QUESTIONS. 

11  What  number  of  dollars  are   in  six  bags,  containing 
each  37542  dollars  ?  Am.  225252. 

2.  If  one  quarter  of  a  ship's  cargo  be  worth  eleven  thou- 
sand and  ninety-nine  dollars,  how  many  dollars  is  the  whole 
cargo  worth  1  -Ans.  44398  dols. 

3.  Money  was  first  made  of  gold  and  silver  at  Argos, 
eight  hundred   and  ninety-four  years  before  Christ  ;  how 
long  has  money  been  in  use  at  this  elate,  1814  1 

Ans.  2708  years. 

4.  The  distance  from  Portland  in  the  Province  of  Maine, 
to  Boston,  is  125  miles  ;  from  Boston  to  New-Haven,  162 
miles  ;  from   thence   to   New-York,  88  ;    from  thence  to 
Philadelphia,  95  ;  from  thence  to  Baltimore,  102  ;  from 
thence  to  Charleston,  South  Carolina,  716 ;  and  from  thence 
to  Savannah,  119  miles — What  is  the  whole  distance  from 
Portland  to  Savannah'?  Ans.  1407  miles. 

5.  John,  Thomas,  and  Harry,  after  counting  their  prize 
money,  John  had  one  thousand  three  hundred  and  seventy- 
live  dollars  ;  Thomas  had  just  three  times  as  many  as  John  ; 
and  Harry  had  just  as  many  as  John  and  Thomas  both — 
Pray  how  many  dollars  had  Harry  1    Ans.  5500  dollars. 


FEDERAL  MONEY. 

NEXT  in  point  of  simplicity,  and  the  nearest  allied  to 
whole  numbers,  is  the  coin  of  the  United  States,  or 

FEDERAL  MONEY. 

This  is  the  most  simple  and  easy  of  all  money — it  in- 
creases in  a  tenfold  proportion,  like  whole  numbers. 
10  mills,  (m.)  make        1  cent,  marked  Cr 

10  cents,  1  dime,  d. 

10  dimes,  1   dollar, 

10  dollars,  1   Baffle. 


ADDITION  OF  FEDERAL  MONEY.  21 

Dollar  is  the  money  unit ;  all  other  denominations  being 
valued  according  to  their  place  from  the  dollar's  place. — 
A  point  or  comma,  called  a  separatrix^  may  be  placed  after 
the  dollars  to  separate  them  from  the  inferior  denominations ; 
then  the  first  figure  at  the  right  of  this  separatrix  is  dimes, 
the  second  figure  cents,  and  the  third  mills.* 


ADDITION  OF  FEDERAL  MONEY. 

RULE. — 1.  Place  the  numbers  according  to  their  value;  that  is, 
dollars  under  dollars,  dimes  under  dimes,  cents  under  cents,  &c.  and 
proceed  exactly  as  in  whole  numbers  ;  then  place  the  separatrix  in 
the  sum  total,  directly  under  the  separating  points  above. 

EXAMPLES. 

d.  c.  m.  $.       d.  c.  m.  $.       d.  c. 


365, 

5 

4 

1 

439, 

3 

0 

4 

136, 

5 

1 

4 

487, 

0 

6 

0 

416, 

3 

9 

0 

125, 

0 

9 

0 

94, 

6 

7 

0 

168, 

9 

3 

4 

200, 

9 

0 

9 

439, 

0 

8 

9 

239, 

0 

6 

0 

304, 

0 

0 

6 

742, 

5 

0 

0 

143, 

0 

0 

5 

111, 

1 

9 

1 

2128,    860 


2.  When  accounts  are  kept  in  dollars  and  cents,  and  no  other  de- 
nomiruuions  are  mentioned,  which  is  the  usual  mode  in  common  reck- 
oning, then  the  first  two  figures  at  the  right  of  the  separatrix  or  point, 
may  be  called  so  many  cents  instead  of  dimes  and  cents  ;  for  the 
place  of  dimes  is  only  the  ten's  place  in  cents  ;  because  ten  cents  make 
a  dime ;  for  example,  48,  75,  forty-eight  dollars,  seven  dimes,  five  cents, 
may  be  read  forty-eight  dollars  and  seventy -five  cents. 

If  the  cents  are  less  than  ten,  place  a  cipher  in  the  ten's  place,  or 
place  of  dimes. — Example.  Write  down  four  dollars  and  7  cents. 
Thus,  #4,  07  cts. 

*  It  may  be  observed,  that  all  the  figures  at  the  left  hand  of  the  separatrix 
are  dollars  ;  or  you  may  call  the  first  figure  dollars,  and  the  other  eagles, 
<fcc.  Thus  any  sum  of  this  money  raay  DC  read  differently,  either  wholly  in 
the  lowest  denomination,  or  partly  in  the  higher,  and  partly  in  the  lowest ; 
for  example,  37  54,  may  be  either  read  3754  cents,  or  375  dimes  and  4 cents, 
or  37  dollars  5  dimes  and  4  cents,  or  3  eagles  7  dollars  5  dimes  and  4  cents, 


ADDITION  OF  FEDERAL  MONEY. 


EXAMPLES. 

1.  Find  the  sum  of  304  dollars,  39  cents ;  291  dollars  9 
cents  ;    136  dollars,  99  cents ;  12  dollars  and  10  cente 
39 


Thus, 


f304, 
I  291,  09 
1  136,  99 
i  12,  10 

Sum,  744,  57  Seven  hundred  forty-four  dol- 
lars and  fifty-seven  cents. 


(3.) 


364, 
21, 

8, 
0, 


cts. 
00 
50 
09 
99 


3287, 

1729, 

4219, 

140, 


cts. 
bO 
19 
99 
01 


(6.) 

$.  cts. 

124,  50 

9,  07, 

0,  60 

231,  01 

0,  75 

24,  00 

9,  44 

0,  95 


8.  What  is  the  sum  total  of  127  dols.  19  cents,  278  dols. 
19  cents,  34  dols.  7  cents,  5  dols.  10  cents,  and  1  dol.  99 
cents?  Ans.  $446,  54  cts. 

9.  What  is  the  sum  of  378  dols.  1  ct.,  136  dols.  91  cts., 
344  dols.  8  cts.,  and  365  dols.  ?  Ans.  $1224. 

10.  What  is  the  sum  of  46  cents,  52  cents,  92  cents,  and 
1.0  cents  ?  ;  Ans.  $2. 

11.  Wliat'"is  the  sum  of  9  dimes,  8  dimes,  and  80  cents  ? 

Ans. 


SIMPLE  SUBTRACTION. 

12.  I  received  of  A,  B,  and  C,  a  sum  of  money;  A  paid 
me  95  dols.  43  cts.,  B  paid  me  just  three  times  as  much  as 
A,  and  C  paid  me  just  as  much  as  A  and  B  both  :  can  you 
tell  me  how  much  inoney  C  paid  me  1    Arts.  $381, 72  cts. 

13.  There  is  an  excellent  well  built  ship  just  returned 
from  the  Indies.     The  ship-only  is  valued  at  12145  dols. 
86  cents ;  and  one  quarter  of  her  cargo  is  worth  25411  dols. 
65  cents.     Pray  what  is  the  value  of  the  whole  ship  and 
cargo*  Ans.  113792,  46  cts. 

A  TAILOR'S  BILL. 

Mr.  James  Paywell, 

To  Timothy  Taylor,  Dr. 

1814,                                             $• cfs-  $• cfs* 
April  15.      To  2£  yds.  of  Cloth,  at  6,  50  per  yd.         16  25 
To  4  yds.  Shalloon,          75 

To  making  your  Coat,  2  50 
To  1  silk  Vest  pattern, 

To  making  your  Vest,  1  50 

To  Silk,  Buttons   &c.  for  Vest,  0  45 

Sum,  $27  80- 

ID*  By  an  act  of  Congress,  all  the  accounts  of  the  United  States, 
thfi  salaries  of  all  officers,  the  revenues,  &c.  are  to  be  reckoned,  in 
sral  money  ;  which  mode  of  reckoning  is  so  simple,  easy,  and  con- 
tent, that  it  will  soon  come  into  common  practice  throughout  all 


ventei 
the  States. 


SIMPLE  SUBTRACTION. 

Subtraction  of  whoh  Numbers, 

TEACHETH  to  take  a  less  number  from  a  greater,  of 
the  same  denomination,  and  thereby  shows  the  difference,* 
or  remainder :  as  4  dollars  subtracted  from  6  dollars,  the  re- 
mainder is  2  dollars. 

RULE. — Place  the  least  number  under  the  greatest,  so  that  units 
may  stand  under  units,  tens  under  tens,  &c.  and  draw  a  line  under 
them. 


SIMPLE  SUBTRACTION. 


2.  Begin  at  the  right  hand,  and  take  each  figure  in  the  lower  line 
from  the  figure  above  it,  and  set  down  the  remainder. 

3.  If  the  lower  figure  is  greater  than  that  above  it,  add  ten  to  the 
upper  figure  ;  from  which  number  so  increased,  take  the  lower  and 
set  down  the  remainder,  carrying  one  to  the  next  lower  number,  with 

'  which  proceed  as  before,  and  so  on  till  the  whole  is  finished. 

PROOF.     Add  the  remainder  to  the  least  number,  and  if  the  sum 
be  equal  to  the  greatest,  the  work  is  right. 


EXAMPLES. 

(1.)  (2.) 

Greatest  number,  2468      62157 
Least  number,        1346      12148 

Difference,  "" 

Proof, 


(4.) 

From  41678839 
Take  31542999 

Rem. 


(5.) 

918764520 
91243806 


(3.) 

8796475 
1  6  4  3  4  8  g 


(6.) 

65432167890 
12345697098 


From     917144043605 
Take       40600S32164 

Rem.  ~ 

(9.)     (10.) 
From  100000   2521665 
Take   65321   2000000 


(8.) 

3562176255002^ 
1235271082165 


(11.) 

200000 

99999 


(12.) 
10000 

I 


Dif.     

13.  From  360418,  take  293752.  Ans.  66666. 

14.  From  765410,  take  34747.  Ans.  730663. 

15.  From  341209,  take  198765.  Ans.  142444. 

16.  From  100046,  take  10009.  Ans.  90037. 

17.  From  2637804,  take  2376982.  Ans.  260822. 

18.  From   ninety  thousand,  five  hundred  and  forty-six 
take  forty-two  thousand,  one  hundred  and  nine. 

Ans.  48437. 

19.  From  fifty-four  thousand  and  twenty-six,  take  nin< 
thousand  two  hundred  and  fifty-four.  Ans.  44772. 


SUBTRACTION  Ol    FEDERAL  MONEY.  Xlr 

20.  From  one  million,  take  nine  hundred  and  ninety-nine 
thousand.  Ans.  One  thousand. 

21.  From  nine  hundred  and  eighty-seven  millions,  take 
nine  hundred  and  eighty-seven  thousand. 

Ans.  986013000. 

22.  Subtract  one  from  a  million,  and  show  the  remainder. 

Ans.  999999. 

QUESTIONS. 

1.  How  much  is  six  hundred  and  sixty-seven  greater 
than  three  hundred  and  ninety-five  1  Ans.  272. 

2.  What  is  the  difference  between  twice  twenty-seven, 
and  three  times  forty-five  I  Ans.  81. 

3.  How  much  is  1200  greater  than  365  and  721  added 
together  1  Ans.  114. 

4.  From  New-London  to  Philadelphia  is  240  miles.   Now 
if  a  man  should  travel  five  days  from  New-London  towards* 
Philadelphia,   at  the  rate  of  39  miles  each  day,  how  far 
would  he  then  be  from  Philadelphia.          Ans.  45  miles. 

5.  What  other  number  with  these  four,  viz.  21,  32,  16, 
and  12,  will  make  100  ?  Ans.  19. 

6.  JPKvine  merchant  bought  721  pipes  of  wine  for  90846 
dollars,  and  sold  543  pipes  thereof  for  89049  dollars  ;  how 
many  pipes  has  he  remaining  or  unsold,  and  what  do  they 
stand  him  in  ? 

Ans.  178  pipes  unsold,  and  they  stand  him  in  $1797. 

SUBTRACTION  OF  FEDERAL  MONEY. 

RULE. — Place  the  numbers  according  to  their  value  ;  that  is,  dollars 
under  dollars,  dimes  under  dimes,  cents  under  cents,  ire.  and  subtract 
as  in  whole  numbers. 

EXAMPLES. 
$.    d.  c.  m. 
From  45,  475 
Take  43,  485 


Rem.  $1,990  one  dollar,  nine  dimes,  and  inn 
*or  one  dollar  and  ninety-nine  < 


OF  l'£D£RA£  MONEY. 


From 
Take 
Kern. 

From 
Take 
Hem. 

From 
Take 
Rern. 

$.      d.  c. 
45,     7  4 
13,     8  9 

$.      d.  c.m. 
46,     246 
36,     1  6  4 

4284 
1993 

$.        cts. 
411,     24 
16,     09 

$.      cts. 
4106,  71 
221,  69 

$.      cts. 
1901,  08 
864,  09 

1 

$.  d.  c.  m, 
211,  110 
111,  1  1  4 


$.  cts. 
960,  00 
136,  41 


$.  cts. 
365,  00 
109,  01 


11.  From  125  dollars,  take  9  dollars  9  cents. 

Ans.  115  dolls.  91  cts. 

12.  From  127  dollars  1  cent,  take  41  dollars  10  cents. 

Ans.  85  dolls.  91  cts. 

13.  From  365  dollars  90  cents,  take  168  dols.  99  cents. 

Ans.  $196,  91  cts> 

14.  From  249  dollars  45  cents,  take  180  dollars.* 

Ans.  $69,  45  cts. 

15.  From  100  dollars,  take  45  cts.     Ans.  $99,  55  cts. 

1 t>.  From  ninety  dollars  and  ten  cents,  take  forty  dollars 
and  nineteen  cents,  Ans.  $49,  91  cts. 

17.  From  forty-one  dollars  eight  cents,  take  one  dollar 
nine  cents.  Ans.  $39,  99  cts. 

18.  From  3  dols.  take  7  cts.  Ans.  $2, 93  cts. 
19  From  ninety-nine  dollars,  take  ninety-nine  cents. 

Ans.  $98,  1  ct. 

;20.  From  twenty  dols.  take  twenty  cents  and  one  milL 

Ans.  $19,  79  cts.  9  mills. 

21.  From  three  dollars,  take  one  hundred  and  ninety-nine 
«'ents.  Ans.  $1,  1  ct. 

22.  From  20  dols.  take  1  dime.          Ans.  $19,  90  cts. 

23.  From  nhie  dollars  and  ninety  cents,  take  ninety-nine 
dimes.  Ans.  0  remains. 

2*4'.  Jack's  prize  money  was  219  dollars,  and  Thomas 


SIMPLE  MULTIPLICATION, 

%•. 

received  just  twice  as  much,  lacking  45  cents.     How  much 
money  did  Thomas  receive  1  Ans.  $437,  55  cts. 

25.  Joe  Careless  received  prize  money  to  the  amount  of 
1000  dollars;  after  which  he  Jays  out  411  dolls.  41  cents 
for  a  span  of  fine  horses  ;  and  123  dollars  40  cents  tor  a 
gold  watch  and  a  suit  of  new  clothes  ;  besides  359  <• 
and  50  cents  he  lost  in  gambling.  How  much  will  he  imv  .• 
left  after  paying  his  landlord's  bill,  which  amounts  to  £•"• 
dojs.  and  11  cents?  Ans.  $20,  58  ct*. 

SIMPLE  MULTIPLICATION 

TEACHETH  to  increase  or  repeat  the  greater  of  tw<  » 
numbers  given,  as  often  as  there  are  units  in  the  less,  or 
multiplying  number  ;  hence  it  performs  the  work  of  man} 
additions  in  the  most  compendious  manner. 

The  number  to  be  multiplied  is  called  the  multiplicand: 
The  number  you  multiply  by,  is  called  the  multiplier. 
The  number  found  from  the  operation,  is  called  the  pro- 
duct. 

NOTE.  Both  multiplier  and  multiplicand  are  in  srenerni 
called  factors,  or  terms. 

CASE  I. 
When  the  multiplier  is  uotiiifore  than  twelve. 

RULE.—  Multiply  each  figure  in  the  multiplicand  by  themul; 
carry  one  for  every  ten,  (as  in  addition  of  whols  numbers,)  and 
will  have  the  product  or  answer. 

PROOF  —  Multiply  the  multiplier  by  the  imiltiplica;- 

EXAMPLES. 

What  number  is  equal  to  3  times  365  ? 

Tbus,     365  multiplier' 

3  multiplier. 


Ans.  IQ95  product. 


*  Multiplication  may  also  be  proved  by  casting  out  the  9's  in  the  two 
factors,  and  setting  down  the  remainders  ;  then  multiplying  the  two  re- 
mainders together ;  if  the  excess  of  9's  in  their  product  is  equal  1o 

•  f  9's  in  the  total  prod.nct,  the  voH,-  H 


SIM  I'LE  M  L-  LTIPLIC  ATJON. 


Multiplicand^ 
Multiplier, 

Product, 

47094 

7 


71085 


5432 
4 


2345  9075 

5  6 


71034 

8 

31261 
9 

4320 
10 

2240613 

4684114 
12 

1432046 
11 


CASE  II. 
When  the  multiplier  consists  of  several  figures. 

RUJ.E. — The  multiplier  being  placed  under  the-  multiplicand,  units 
under  units,  tens  under  tens,  &c.  multiply  by  each  significant  figure 
in  the  multiplier  separately,  placing  the  first  figure  in  each  product 
exactly  under  its  multiplier  ;  then  add  the  several  products  together 
in  the  same  order  as  they  stand,  and  their  sum  will  be  the  total  product, 
EXAMPLES. 

What  number  is  equal  to  47  times  365  ? 

Multiplicand,    365 
Multiplier,  4  7 


2555 
460 


1 


Ans.   17155  product. 


Multiplicand,  37864 
Multiplier,      209 

340776 

75726 


Prodi 


7913576 


34293 
74 


2537682 


47042 
91 


820 
6816978 


25203 
4025 


4280822 

2193 
4072 


9876 
9405 


101442075 


8929896    92883780 


269181 
4629 


SlMl'LK  Ml' 

;>61986 


42068 


1246038849 


134092 
87362 


2001049068     1709391112 

918273645 
1003245 


11714545304 


921253442978025 


14.  Multipl^r604S3  by  915;2  Ans.  6959940416. 

15    Wh^  ;    4l        »tal  product  01  7608  times  <ft>54d 

Ans.  278020665(5. 
16.  What  number  is  equal  to  40003  times 


CASE  III. 

When  there  are  ciphers  on  the  right  hand  of  either  or 
both  of  the  factors,  neglect  those  ciphers  ;  then  place 
significant  figures  under  one  another,  and  multiply  by  tJ 
only,  and  to  the  right  hand  of  the  product,  place  as  mai 
ciphers  as  were  omitted  in  both  the  factors. 


21200 
70 

1484000 


EXAMPLES, 

31800 
36 

1 144800 


84600 
34000 

2876400000 


3040 


109215040000 


98260000 
8109397800000 


7065000  X  8700=61465500000 

749643000  x  695000^521001885000000 

360000  x  1200000^432000000000 

CASE  IV. 

When  the  multiplier  is  a  composite  number,  that  is,  when 
it  is  produced  by  multiplying  any  two  numbers  in  the  ta 
together;  multiply  first  by  one  of  those  fiffin 


30  [MPIrE  MULTIPLICATION. 

product  by  the  other,  and  the  last  product  will  he  the  total 
required. 

EXAMPLI 

Multiply  41364  by  35. 
7x5=35.  7 


289548  Product  of 


,__  1447740  Product  of  35. 



2.  Multiply  764131  by  48.  Ans.  36678288. 

3.  Multiply  3425)6  by  56.  Ans.  19180896. 

4.  Multiply  209402  by  72.  Ans.  15076944. 

5.  Multiply  91738    by  81.  Ans.  7430778. 

6.  Multiply  34462    by  108.  Ans.  3721896. 

7.  Multiply  615243  by  144.  Ans.  88594992. 

CASE  V. 

To  multiply  by  10,  100,  1000,  &c.  annex  to  the  multi- 
plicand all  the  ciphers  in  the  multiplier,  and  it  will  make 
the  product  required. 

EXAMPLES. 

1.  Multiply  365      by  10.  Ans.  3650. 

2.  Multiply  4657    by  100.  Ans.  465700. 

3.  Multiply  5224    by  1000.  Ans.  5224000. 

4.  Multiply  2646Q  by  10000.  Ans.  264600000. 

EXAMPLES   FOR  EXERCISE. 

1.  Multiply  1203450  by  9004.  Ans.  10835863800. 

2.  Multiply  9087061  by  56708.  Ans.  515309055188. 

3.  Multiply  8706544  by  67089.  Ans.  584113330416. 

4.  Multiply  4321209  by  123409.  Ans.  533276081481. 

5.  Multiply  3456789  by  567090.  Ans.  1960310474010. 

6.  Multiply  8496427  by  874359.  Ans.  7428927415293. 

98763542  x  98763542—8754237228383764. 

Application  and  Use  ^f  Multiplication. 
In  making  out  bills  qf  parcels,  and  in  finding  the  value  of 
goods  ;  when  the  price  of  one  yard,  pound,  &c.  is  given  (in 
Federal  Money)  to  find  the  value -of  the  whole  quantity. 


SIMPLE   MULTIPLK  ATIO.V.  31 

RULE. Multiply   the  given   price  and   quantity  together,  as  in 

whole  numbers,  and  the  separatrix  will  be  as  many  figures   from  the 
right  hand  in  the  product  as  in  the  given  price. 
EXAMPLES. 

1.  What  will  35  yards  of  broad-  >  $.  d.  c.  m. 
cloth  come  to,  at  J  3,  4  9  6  per  yard  ? 

3  5 


17  4  8  0 

104  8  8 

Ans.  $122,  3  6  0=122  dol- 
[lars,  36  cents. 

2.  What  cost  35  Ih.  cheese  at  8  cents  per  Ib.  1 

,08 

Ans.  $2,  80—2  dollars  80  cents. 

3.  What  is  the  value  of  29  pairs  of  men's  shoes,  at  1  dol- 
lar 51  cents  per  pair?  Ans.  $43,  79  cents. 

4.  What  cost   131  yards  of  Irish  linen,  at  38  cents  per 
yard  ?  Ans.  $49,  78. cents. 

5.  What  cost  140  reams  of  paper,  at  2  dollars  35  cent- 
per  ream  ?  Ans.  $329. 

6.  What  cost  144  Ib.  of  hyson  tea,  at  3  dollars  51  cents 
perlb.  1  Ans.  $505,  44  cents. 

7.  What  cost  94  bushels  of  oats,  at  33  cents  per  bushel  ? 

Ans.  $31,  2  cents. 

8.  What  do  50  firkins  of  butter  come  to,  at  7  dollars  14 
cents  per  firkin  ?  Ans.  $357. 

9.  What  cost  12  cwt.  of  Malaga  raisins,  nt  7  dollars  31 
cents  per  cwt.  ?  Ans,  $87,  72  cents. 

10.  Bought  37  horses  for  shipping,  at  52  dollars  per  head  : 
what  do  they  come  to  ?  Ans.  $1924. 

11.  What  is  the  emoiiT>t  of  500  Ibs.  of  IKo.o-V-larcl,  at  15 
cents  per  Ib.  ?  ,   Ans.  $75. 

12.  What  is  the  value  of  75  yards  of  «^tin,  at  3  dollars 
75  cents  per  yard  ?  Ans.  $281 , 25. 

13.  What  cost  307  acre*  of  land,  at  14  dols.  67   cents 
per  acre  ?  .4??«?.  45383,  89  cei 


32  DIVISION  OF  WHOLE  NUMBERS. 

14.  What  does  857  bis.  pork  come  to,  at  18  dols.  93 
cents  per  bl.  ?  Am.  $16223,  1  cent. 

15.  What  does  15  tuns  of  hay  come  to,  at  20  dols.  78 
cts.  per  tun  ?  Ans.  $311,  70  cents. 

16.  Find  the  amount  of  the  following 

BILL  OF  PARCELS. 

New-London,  March  9,  1814. 

Mr.  James  Paywell,  Bought  of  William  Merchant. 

$.  cts. 

28  Ib.  of  Green  Tea,  at  2,  ISperlb. 
41  Ib.  of  Coffee,                          at  0,  21 

34  Ib.  of  Loaf  Sugar,  at  0,  19 

13  cwt.  of  Malaga  Raisins,  at  7,  31  per  cwt. 

35  firkins  o(1  V-utter,  at  7,  14  per  fir. 
27  pairs  oi  worsted  Hose,  at  1,  04  per  pair. 
94  bushels  of  Oats,  at  0,  33  per  bush. 

29  pairs  of  men's  Shoes,  at  1,  12  per  pair. 

Amount,  $511,  78. 
Received  payment  in  full,  WILLIAM  MERCHANT. 

A  SHORT  RULE. 

NOTE.     The  value  of  lOOlbs.  of  any  article  will  be  just 
as  many  dollars  as  the  article  is  cents  a  pound. 
For  100  Ib.  at  1  cent  per  lb.=100  cents— 1  dollar. 
100  Ib.  of  beef  at  4  cents  a  Ib.  comes  to  400  cents=4 
dollars,  &c. 


DIVISION  OF  WHOLE  NUMBERS. 

SIMPLE  DIVISION  teaches  to  find  how  many  times 
on*  whole  number  is  contained  in  another  ;  and  also  what 
remains  ;  and  is  a  concise  way  of  performing  several  sub- 
tractions. 

Four  principal  parts  are  to  be  noticed  in  Division  : 

1.  The  Dividend,  or  number  given  to  be  divided. 

2.  The  Divisor,  or  number  given  to  divide  by. 

3.  The  Quotient,  or  answer  to  the  question,  which  shows 
how  many  times  the  divisor  is  contained  in  the  dividend. 

4.  The  Remainder,  which  is  always  less  than  the  divisor, 
and  of  the  same  name  with  the  Dividend. 


DIVISION  OF  WHOLE  LUMBERS. 

LE. — First,  seek  how  many  times  the  divisor  is  contained  in  as 
many  of  the  left  hand  figures  of  the  dividend  as  are  just  necessary  ; 
(that  is,  find  the  greatest  figure  that  the  divisor  can  be  multiplied  by, 
so  as  to  produce  a  product  that  shall  not  exceed  the  part  of  the  divi- 
dend used ;)  when  found,  place  the  figure  in  the  quotient  ;  multiply 
the  divisor  by  this  quotient  figure  ;  place  the  product  under  that  part 
of  the  dividend  used  ;  then  subtract  it  therefrom,  and  bring  down  the 
next  figure  of  the  dividend  to  the  right  hand  of  the  remainder  ;  after 
which,  you  must  seek,  multiply  and  subtract,  till  you  have  brought 
down  every  figure  of  the  dividend. 

PROOF.  Multiply  the  divisor  and  quotient  together,  and  add  the 
remainder,  if  there  be  any,  to  the  product ;  if  the  work  be  right,  the 
sum  will  be  equal  to  the  dividend.* 

EXAMPLES. 

1.  How  many  times  is  4  2.  Divide  3656  dollars 

contained  in  9391 1  equally  among  8  men. 

Divisor,  Div.  Quotient.  Divisor,  Div.  Quotient, 
4)9391(2347  8)3656(457 

8  4  32 


13       9388  45 

12         +3  Rem.  40 


19     9391  Proof.  56 

16  56 


31  3656  Proof  by 

28  addition. 

3  Remainder. 


*  Another  method  which  some  make  use  of  to  prove  division  is  as  fol- 
lows :  viz.  Add  the  remainder  and  all  the  products  of  the  several  quotient 
figures  multiplied  by  the  divisor  together,  according  to  the  order  in  which 
they  stand  in  the  work  ;  and  this  sum,  when  the  work  is  right,  will  be  equal 
to  the  dividend. 

A  third  method  of  proof  by  excess  of  nines  is  as  follows,  viz. 

1.  Cast  the  nines  out  of  the  divisor,  and  place  the  excess  on  the  left,  hand. 

2.  Do  the  same  with  the  quotient,  and  place  it  on  the  right  hand. 

3.  Multiply  these- two  figures  together,  and  add  their  product  to  the  re- 
mainder, and  reject  the  nines,  and  place  the  excess  at  top. 

4.  Oast  the  nines  out  of  the  dividend,  and  place  the  excess  at  bottom. 
«Vo/f.     If  the  sum  is  right,  (he  top  and  bottom  figures  will  be  aiiko. 


\ 


0»4  DIVISION  OF  WHOLE  LUMBERS. 

Divisor.  Div.  Quotient. 

29)15359(529  365)49640(136 

145  365 

Proof  by        

excess  of  9's.           85  1314 

5                       58  1095 


279  2190 

261  2190 


Remains    18  0  Rem. 

v     Divisor.  Div.  Quotient.  95(85595(901 

61)28609(469  736)863256(1172 

472)251 104(532  there  remains  664. 

9.  Divide  1893312  by  912.  Ans.  2076. 

10.  Divide  1893312  by  2076.  Ans.  912. 

11.  Divide  47254149  by  4674.  Ans.  10110  T/T7. 

12.  What  is  the  quotient  of  330098048  divided  by  4207  ? 

Ans.  78464. 

13.  What  is  the  quotient  of  761858465  divided  by  8465  ?  . 

Ans.  90001. 

14.  How  often  does  761858465  contain  90001  ? 

Ans.  8465. 

15.  How  many  times  38473  can  you  have  in  119184693  ? 

Ans.  3097f|fif. 

16.  Divide  280208122081  by  912314. 

Quotient,  307140yTVVrT- 

MORE  EXAMPLES  FOR  EXERCISE. 

Divisor.  Dividend.  Remainder. 

234063)590624922(  Quotient)S3973 
47614)327879186(  )  9182 

987654(988641654(  )  -  -  -  0 

CASE  II. 

When  there  are  ciphers  at  the  right  hand  of  the  divisor, 
cut  off  the  ciphers  in  the  divisor,  and  the  same  number  of 
figures  from  the  right  hand  of  the  dividend  ;  then  divide  the 
remaining  ones  as  usual,  and  to  the  remainder  (if  any)  an- 
nex  those  figures  cut  off  from  the  dividend,  and  you  will 
have  the  true  remainder. 


DIVISION  OF   WHOLE  .NUMBERS.  35 

EXAMPLES. 

1.  Divide  4673625  by  21400. 

w  true  quotient  by  Restitution. 

428-- 

~393 
214 

1796 
1712 


8425  true  rein. 

2.  Divide  379432675  by  6500.  Ans.  58374£f |f . 

3.  Divide  421400000  by  49000.  Ans.  8600. 

4.  Divide  11659112    by  89000.  Ans.    131¥^|7. 

5.  Divide  9187642      by  9170000.  Ans.     M^ffo. 

MORE  EXAMPLES. 

Divisor.  Dividend.  Remains. 

125000)436250000(  Quotient.  )  0 

120000)  149596478(  )  76478 

901000)654347230(  )221230 

720000)987654000(  )534000 

CASE  III. 

Short  Division  is  when  the  Divisor  does  not  exceed  12. 
RULE. — Consider  how  many  times  the  divisor  is  contained  in  the 
first  figure  or  figures  of  the  dividend,  put  the  result  under,  and  carry 
as  many  tens  to  the  next  figure  as  there  are  ones  over. 

Divide  every  figure  in  the  same  manner  till  the  whole  is  finished. 

EXAMPLES. 
Divisor.  Dividend. 

2)113415          3)85494          4)39407        5)94379 


Quotient,  56707—1 


6)120616  7)152715  8)96872  9)118724 


11)6986197  12)14814096  12)570196382 


CONTRACTIONS  IN  DIVISION. 

Contractions  in  Division. 

Whea  the  divisor  is  such  a  number,  that  any  two  figures 
in  the  Table,  being  multiplied  together,  will  produce  it,  di- 
vide the  given  dividend  by  one  of  those  figures  ;  the  quo- 
tient thence  arising  by  the  other  ;  and  the  last  quotient  will 
be  the  answer. 

NOTE.  The  total  remainder  is  found  by  multiplying  the 
last  remainder  by  the  first  divisor,  and  adding  in  the  first 
remainder. 


EXAMPLEvS. 


Divide  162641  by  72 


9)162641           or 

8)162641 

last  rem..  7 
X9 

8)18071—2 

9)20330—1 

2258—7 

£258—8 

63 

first  rcm.  +2 

True  rem.  65 
Ans.  11154. 
Ans.  19475if  . 
Ans.  26924^- 

Ans.  2212^- 
Ans.  3018TV 
Ans.  17359^- 
Ans.  1118ff. 
Ans.  1345. 
Ans.  10940. 

True  Quotient  2258ff. 

2.  Divide  178464    by  16. 
3.  Divide  467412    by  24. 
4.  Divide  942341    by  35. 
5.  Divide    79638    by  36. 
6.  Divide  144872    by  48. 
7.  Divide  937387    by  54. 
8.  Divide    93975    by  84. 
9.  Divide  145260    bv  108. 
10.  Divide  1575360  by  144. 

2.  To  divide  by  10,  100,  1000,  &c. 

RULE. — Cutoff  as  many  figures  from  the  right  hand  of  the  dividend 
as  there  are  ciphers  in  the  divisor,  and  these  figures  so  cut  off  arc  the 
remainder ;  and  the  other  figures  of  the  dividend  are  the  quotient. 

EXAMPLES. 


1.  Divide  365         by  10. 

2.  Divide  5762       by  100. 

3.  Divide  763753  by  1000. 


Ans.  36  and  5  remains. 
Ans.  57  —  62  rcm. 
Ans.  763  —  753  rem. 


SUPPLEMENT  MY;  MULTIPLICATION,  3T 

SUPPLEMENT  TO  MULTIPLICATION. 

To  multiply  by  a  rnixt  number  ;  that  is,  a  whole  number 
joined  with  a  fraction,  as  8|,  5i,  6J,  &c. 

RULE. — Multiply   by  the  whole  number,  and  take  •$,  i,  f,  &c.    of 
the  multiplicand,  and  add  it  to  the  product. 
EXAMPLES. 

Multiply  37  by  23i.  Multiply  48  by  2J. 

2)37  48 


18i 
74 

8691 

3.  Multiply 

4.  Multiply 

5.  Multiply 

6.  Multiply 


Answer. 

211  by 
2464  by 

345  by 
6497  by 


50i. 


96 

132  Ans. 

Ans.  106551. 

Ans.  20533  J. 

Ans.  6598-1. 

Ans.  334131. 


Questions  to  exercise  Multiplication  and  Division. 

1.  What  will  9|  tuns  of  hay  come  to,  at  14  dollars   a 
tun  ?  '_4n*.-$136£. 

2.  If  it  take  320  rods  to  make  a   mile,  and  every  rod 
contains  51  yards ;  how  many  yards  are  there  in  a  mile  ? 

Ans.  1760. 

3.  Sold  a  ship  for  11516  dollars,  and  I  owned  f  of  her ; 
what  was  my  part  of  the  money  1  Ans.  $8637. 

4.  In  276   barrels  of  raisins,   each  3i  cwt.  how   many 
hundred  weight  ?  Ans.  966  cwt. 

5.  In   36  pieces    of  cloth,  each   piece  contain!  n- 
yards  ;  how  many  yards  in  the  whole  ?         Ans.  873  yds. 

6.  What  is  the  product  of  161  multiplied  by  itself? 

Ans.  25921. 

7.  If  a  man   spend   492  dollars  a  year,  what  is  that  per 
calendar  month  ?  Ans.  $41. 

8.  A    privateer  of  65  men   took  a  piize,   which   being 
equally  divided  among  them,  amounted  to  H9/.  per  man  : 
what  is  the  value  of  the  prize  ? 


COMPOUND  ADD1TIOA. 

9.  What  number  multiplied  by  9,  will  make  225 1 

Am.  25. 

10.  The    quotient  of  a  certain  number  is  457,  and  the 
divisor  8  ;  whr?t  is  the  dividend  1  Ans.  3656. 

11.  -What  cost  9  yards  of  cloth,  at  3s.  per  yard  1 

Ans.  27s. 

12.  What  cost  45  oxen,  at  8J.  per  head  1      Ans.  £360. 

13.  What  cost   144  Ib.  of  indigo,  at  2  dols.  50  cts.  or 
2-SO  cents  per  Ib.  Ans.  $360. 

14.  Write  down  four  thousand  six  hundred  and  seven- 
teen, multiply  it  by  twelve,  divide  the  product  by  nine,  and 
add  365  to  the  quotient,  then  from  that  sum  subtract  five 
thousand  five  hundred  and  twenty-one,  and  the  remainder 
will  be  just  1000.     Try  it  and  see. 


COMPOUND  ADDITION, 

IS  the  adding  of  several  numbers  together,  having  dif- 
ferent denominations,  but  of  the  same  generic  kind,  as 
pounds,  shillings  and  pence,  &c.  Tuns,  hundreds,  quar- 
ters, &c. 

RULE. — 1.  Place  the  numbers  so  that  those  of  the  same  denomina- 
tion may  stand  directly  under  each  other. 

2.  Add-the  first  column  or  denomination  together,  as  in  whole  num- 
bers ;  then  divide  the  sum  by  as  many  of  the  san-o  denomination  as 
make  one  of  the  next  gn.-ater ;  setting-  down  the  remainder  under  the 
column  odded,  and  carry  the  quotient  to  *he  next  superior  denomina- 
tion, continuing  the  same  to  the  last,  which  add,  as  in  simple  addition.* 

1.  STERLING  MONEY, 

Is  the  money  of  account  in  Great-Britain,  and  is  reckon- 
ed in  Pounds,  Shillings,  Pence  and  Farthings.  See  the 
Pence  Tables. 

*  The  reason  of  this  rule  is  evident :  For,  addition  of  this  money,  as  1 
in  the-  pence  is  equal  to  4  in  the  farthings  j  1  in  the  shillings,  to  12  in  the 
pence  ;  and  1  in  the  pounds,  to  20  in  the  shillings  ;  therefore  currying  as  di- 
rected, is  the  arranging  the  monov,  arising  from  each  column  properly  in 
the  scale  of  denominations  :  and  this  reasoning  will  hold  good  mthe  ad- 
.ditionof  compound  numbers  of  any  denomination  whatever. 


COMPOUND  ADDITION. 


EXAMPLES.                             £•        S.         d. 

What  isfthe  sum  total  of  47/.  13s.            C  47     13       6 
6d.~19/.  2s.  9id.-—  14?.   10s.   ll\d.  r™      j  19      2       9i 
and  12/.  9s.  l$d.                                         IS1  14     10     ll| 
1  12       9       If 

Answer,  £.93     16       4J 

(2.)                          (3.) 
£.     s.      d.           £.      s.      d.  qr. 
17    13    11          84    17      53 
13    10      2          75    13      4    3 
10    17      3          50    17      8    2 
87          20    10     10    1 
334                 16      5  ,0 

(4.) 
£.     s.      d.  qr. 
30    11     4    2 
15     10    9    1 
1011 
3983 
4631 

• 

(5.) 
£.      5.       d.  qr. 
47     17       6k; 
3      9     10    3 
59    17     11     2 
317     16      9    3 
762    19     10    1 
407    17      6    2 
1     19      9    0 

(6.) 
£.     s.      d.  qr. 
7     17     10    3 
60      6       8    0 
7     14     11     2 
18     19      9    3 
91     15      82 
18     17     10    3 
5012 

(?•)• 
£.        s.      d.  qr. 
541       000 
711      9      8     1 
918      6      9    3 
140    15    10    1 
300     19     11     3 
48     10      73 
0     14      9    3 

(8.) 
£.     s.       d. 
105     17      6 
193     10     11 
901     13      0 
319    19      7 
48     17      4 
104    11       9 
96    16      7 
111      9      9 
976      0     10 
449     12      6 
29    10      4 

(9.) 
£.      s.     d. 
940     10      7 
36      9     11 
11       4     10 
141     10      6 
126     14      0 
104     19      7 
!(>{)     10      6 
100      0      0 
909 
0    19      6 
120      0      8 

(10.) 
£.      s.       d. 
97     11       6A 
20       0       4 
144       1     10 
17     11 
9     16     1(H 
0     19      9-1 
19      9      4 
234     11     10* 
180     14      6 
421     10      3i 
341     10      4 

sum 

17s.  8rf.—  137.  Os.  7d.—  19s. 

and  15/,  6s. 


11.  Find  the  amount  of  the  following^         £. 
ms,  vb.  4ftl.  13s.  5,/.—  1U  105.—  41.  \ 

.—27L  ( 

3 


Ans.  £.    115     7     0.T 


40  COMPOUND  ADDITION. 


Add  304Z.  5s.  and 
Os.  lid.—  19s.  6d.  Iqr.  and  45/.  together. 

Ans.  £.  640  3s.  5J-rf. 

13.  Find  the  sum  total  of   14/.  19s.  6d.~llL  4s.  9<7. 
25/.  105.—  4/.  Os.  6d.—3l.  5s.  8d.—  19*.  6<£  and  Os.  6d. 

Ans.  £.  60  Os.  5d. 

14.  Find  the  amount  of  the  following  sums,  viz. 
Forty  pounds,  nine  shillings,  -     -     -     -     -    £.     s.     d. 
Sixty-four  pounds  and  nine  pence,          -     - 
Ninety-five  pounds,  nineteen  shillings,    - 

Seventeen  shillings  and 


Ans.  £.  201  6s.  Ud. 


15.  How  much  is  the  sum  of 
Thirty -seven  shillings  and  sixpence,  - 
Thirty-nine  shillings  and  4jr/.    -  -  -  -     ' 
Forty-four  shillings  and  nine  pence,    - 
Twenty-nine  shillings  and  three  pence, 
Fifty  shillings, 


Ans.  £.  10  Os. 

16.  Bought  a  quantity  of  goods  for  125/.   10s. ;  paid  for 
truckage,  forty-five  shillings,  for  freight,  seventy-nine  shil- 
Mngs  and  sixpence,  for  duties,  thirty-five  shillings  and  ten 

ence,  andTny  expenses  were  fifty-three  shillings  and  nine 
>ence  ;  what  did  the  goods  stand  me  in  1 

Ans.  £.  136  4s,  Id. 

17.  Six  men  took  a  prize,  and  having  divided  it  equally 
amongst  them,  each   man  shared  two  hundred  and   forty 
pounds,    thirteen  shillings  and    seven  pence  ;  how  much 
money  did  the  whole  prize  amount  to  ? 

Ans.  £.  1444  Is.  6d. 

2.    TROY  WEIGHT. 

//;.  oz.  pwt.  gr.  Ib.     oz.   pwt.  gr. 

J(i  11  19  23  8  11  19  21 

4  4  16  21  0  10  16  8 

8  8  19  14  7  8  17  21 

0  9  14  17  468  23 

4  7  10  7  9  7  14  17 

0  7  11  12  7  9  13  10 


POUND  ADDITION.  41 

3.    AVOIRDUPOIS  WEIGHT. 

cwt.  qr.  lb.  lb.    oz.    dr.  T.  cwt.  qr.  lb.    oz.    dr. 

2  3    27  24    13    14  91     17    2    24    13    14 
1     1     17 

4  2  26 

6  1  13 

3  3  15 
6  2  16 


3  9  gr. 
9  1     17 
329 
6  1     17 

4  0    16 

5  2    12 

6  1     10 


17 

12 

11 

19 

9 

0 

17 

10 

12 

26 

12 

15 

14 

13 

2 

04 

9 

11 

16 

8 

7 

47 

11 

3 

]9 

J4 

5 

24 

10 

12 

69 

00 

1 

00 

00 

12 

11 

12 

12 

77 

19 

3 

27 

15 

1J 

^RIES 

4.    APOTHEGM 

WEIGHT. 

g 

3 

9 

gr. 

fe 

3 

3 

9 

gr. 

10 

7 

2 

19 

12 

11 

6 

1 

15 

6 

3 

0 

12 

4 

9 

7 

0 

12 

7 

6 

1 

7 

9 

10 

I 

2 

16 

9 

5 

^j 

12 

j 

8 

1 

2 

19 

6 

1 

0 

16 

9 

0 

0 

1 

10 

9 

;> 

t> 

19 

4 

9 

2 

1 

6 

5.    CLOTH  MEASURE. 

yd.  qr.  na.  E.  E.  qr.  na.                       E.  F.  qr.  na. 

71    3    3  44    3  2  84  2  1 

13    2    1  49    4  3  07  1  3 

10    0    1  06    2  3  76  0  2 

42    3    3  84    4  1  52  2  3 

57    2    2  07    0  0  53  2  2 

49    2    2  61    2  1  09  2  3 


G.    PhY   MEASURE. 

?k.qt.pt.  lu.  pk.  at.  lu.  vie.  qt.  pt. 

7    1  17 '2  V  2i>'3  7    1 

260  34    27  64    261 
150  13    3    6  43    0    4    0 
241  16    3    4  52    3    5    1 

261  27    26  94    230 
360  56    07  54a70 


7.    WINE   MEASURE. 

gal.  at.  pt.  gi.  hhd.  gal  qt.  pt.  tun.hhd.  gal.  qt. 

39  3  1  ^  42  61  31  34  2  %4  2 
17  2  1  2  27  39  2  0  19  1  59  J 
24  3  0  I  9  14  0  1  28  2   2  1 
19  1  1  2  0921  19  0  32  2 

8003  16    24    1     1  37    3    11     1 

40  2    I    1  5    00    3    0  0190 


42  '  K)U  N  D   ADD!  T  i » >  \  . 

8.    LONG  MEASURE. 


yds.  ft.  in.  b.c. 
'4    '2    11    2 

in.  fur. 
46    4 

po. 
16 

le.   m. 

86    2 

fur.  pu. 
6    32 

3 

1       8    1 

58    5 

23 

52    1 

7 

16 

1 

292 

9    6 

34 

64    2 

5 

19 

6 

2    10    1 

17    4 

18 

73    1 

4 

15 

1 

061 

7    3 

15 

7    2 

3 

25 

3 

170 

5    2 

24 

28    2 

4 

17 

9.    LAND 

OR  SQUARE  MEASURE. 

(icres.  roods,  rods. 

acres. 

,   roods 

.  rods 

sg. 

ft- 

sg.  in. 

478 

3 

31 

856 

2 

18 

5 

136 

816 

2 

17 

19 

3 

00 

6 

129 

49 

1 

27 

9 

1 

39 

8 

134 

63 

3 

34 

1 

3 

00 

0 

143 

9 

3 

37 

0 

2 

27 

4 

34 

SOLID 

10. 

MEASURE. 

T. 

ft- 

cords. 

/fee*. 

feet. 

inches. 

41 

43 

3 

122 

13 

1446 

12 

43 

4 

114 

16 

1726 

49 

6 

7 

8'} 

3 

866 

-4 

27 

10 

127 

14 

284 

11.    TIME. 

Y.     m. 

w.  da. 

Yr. 

da. 

&.     w. 

sec. 

57    11 

3    6 

24 

363 

23    54 

34 

3      9 

2    3 

21 

40 

12    40 

24 

29      8 

o      rr 

13 

112 

14    00 

17 

46    10 

2    4 

14 

9 

11     18 

14 

10      7 

1    2 

8 

24 

8     16 

13 

12.    CIRCULAR  MOTION. 

«      o         in                            o        o         /  // 

3*  29  17  14        11  29  59  50 

1   6  10  17         0  00  40  10 

4  18  17  11         94  10  49 

6  14  18  10         4  11   6  10 


COMPOUND  SUBTRACTION. 


COMPOUND  SUBTRACTION, 

TEACHES  to  find  the  difference,  inequality,  or  excess, 
between  air.  two  sums  of  diverse  denominations. 

RUL-:. — Place  those  numbersxunder  each  other,  which  are  of  the 
same  denomination,  the  less  being  below  the  greater  ;  begin  with  the 
least  denomination,  and  if  it  exceed  the  figure  over  it,  borrow  as  many 
units  us  make  one  of  the  next  greater  ;  subtract  it  therefrom  ;  and  to 
the  difference  add  the  'ipper  figure,  remembering  always  to  add  one 
to  the  next  superior  d-  ,  omination  for  that  which  you  borrowed. 

NOTE.  The  method  of  proof  is  the  same  as  in  simple  subtraction, 
EXAMPLES. 

Sterling  Money. 

'(2.)  (3.) 

£.    s.  d.  qr.  £.    s.  d. 

14  14  6  2  94  11  6 

10  19  6  3  36  14  8 


1. 

£.     s!  d.qr. 
From    346  16  5  3 
Take    128  17  4  2 


Rem.    217  19  1  1 


(4.) 

£.  s.    d. 

Borrowed    44  10  2 
Paid  93  11  8 


Remains 
unpaid 


Lent 
Received 

Due  to  me 


(5.) 

£.    s.  d,  qr. 
36     082 
18  10  7  3 


(6.) 

£.  s.  d. 
From  500 
Take  4  19  11 


Rem. 


From 
Take 

Rem. 


(9.) 

£.     s.     </;  or. 
141   14     9  2 
19  13  10  2 


£.  s.    d.  qr. 
7  11   1  2 
4  17  3  1 


£.  s.  d. 
125  01  8 
124  19  8 


(8.) 

£.     s.  d.qr. 
476  10  9  1 

277  17  7  1 


(11.) 

£.    s.  d.  yr. 

10  13  7  1 

0963 


44  COMPOUND  SUBTRACTION. 

12.  Borrowed  27/.  11s.  and  paid  19Z.  17s.  M.  how  much 
remains  due  1  Ans.  £7  13s.  6d. 

43.  How  much  does  317/.  6s.  exceed  1787.  18s.  5^7.  ? 

Ans.  £13«  7s.  6%d. 

14.  From  eleven  pounds  take  eleven  pence. 

Ans.  £10  19s.  Id. 

15.  From  seven  thousand  two  hundred  pounds,  take  18Z. 
17s.  6±d.  Ans.  £7181  2s.  5\d. 

16.  How   much   does  seven  hundred   and  eight  pounds, 
exceed  thirty-nine  pounds,  fifteen  shillings  and  ten   pence 
halfpenny  ?  Ans.  £668  4s.  l\:d. 

17.  From   one  hundred   pounds,   take  four  pence  'half- 
penny. Ans.  £99  19s.  7{d. 

18.  Received  of  four  iner  the  following  sums  of  money, 
viz.     The  first  paid  me  377.  11s.  ±<i.  me  second  257.  16s. 
7d.  the   third  19/.  14s.  tid.  and  the  fourth   as  much   as   all 
the  other  three,  lacking  19s.  6d.     I  demand  the  wl,o!«>  sum 
received  ?  Ans.  £165  5s.  irf. 

2.    TROY  WEIGHT. 

Ib.  oz.  pwt.  oz.  pwt.  fir.  Ib.  oz.  piut.  p*r. 
From  6  11  14  4  19  21  44  9  6  12 
Take  2  3  16  2  14  23  17  3  16  18 


Rem. 


Ib.  oz.  pwt.  gr.  Ib.  oz.  pwt.gr. 

684  2  10  14          942  200 
683  1   9  13          892  9  2  3  . 


3.    AVOIRDUPOIS  WEIGHT. 

Ib.  oz.    dr.             cwt.  or.    Ib.                T.  cwt.  or.  Ib.  oz.  dr. 

7      9     12                 7    3     13                7     10     3  17  5  12 

3     12      9                5     1     15                3     12     1  19  10  9 


T.  cwt.  qr.  Ib.  oz.  Ir.  T.  cwt.  or.  Ib.  oz.  dr. 
810  11  0  20  10  11  317  12  I  12  9  12 
193  17  1  20  12  14  180  12  1  14  10  14 


COMPOUND  SUBTRACTION.  45 

4.  APOTHECARIES'  WEIGHT. 


fc      3   3 
19      8    7 
9    11     6 

3    B   gr. 
4     1     17 
1     2     15 

ft     g     3     9  gr. 
35    7     3     1     14 
17  1U    6     1     18 

Yd.  qr.  na. 
35     1     2 
19     1     3 

5.    CLOTH   MEASURE. 

E.E.    qr.  na. 
467    3    1 
291     3    2 

E.Fl.  qr.  na. 
765     1     3 

149     2     1 

.E.l^.  qr.  na. 

845     1     1 
576    2    3 

Yd.  qr.  na. 
813    3    1 
174    1     0 

E.E.  qr.  na. 
615     0     1 
226    2    2 

bu.  pk.  qt. 
65     1     7 
14    3    4 

6.    DRY  MEASURE. 

bu.  pk.  qt. 
8     1     5 
316 

17    23    0 
6261 

ff«*l- 

14    2    1     3 

7.    WINE  MEASURE. 
hhd.  gal.  qt.  pt. 
13      0     1     0 
10    60    3     1 

T.  hhd.  zal.  qt.  pt. 
2    3    20    3    1 
1     2    27    0    0 

hhd.  gal. 
612     23 
75  '37 

qt.  pt.                         hhd. 
1     0                           521 
1     1                         256 

gal.  qt.  pt. 
14    \     I 
25    3    0 

yd.  ft.  in.   b.c. 
4    2     11     0 
2    2    11     1 

8.    LONG  MEASURE. 
m.  fur.  po. 
41     6    22 
10     6    23 

le.    m.  fur.  po. 
86    2    6     32 
24     1     7    31 

le.     in.  fur,po. 
27    1    6    37 
19    2    4    39 

le.  m.  fur.  po. 
16     U^    i     3 
10     1     3    5 

fo.  m.  fur.  po. 
9    2^0    7 
1118 

46 


COMPOUND  SUBTRACTION. 


9.  LAND  OR  SQUARE  MEASURE. 

A.     roods,  rods.  A.    r.  po. 

29        1        10  29    2    17 

24        1        25  17    1    36 


or.  r 
540  0  25 
119  1  27 


tuns.  ft. 
116  24 
109  39 


A.   or.  rods. 
130   1   10 
49   1   11 


10.    SOLID  MEASURE. 

cords,  ft. 
72  114 
41  120 


19 


143 


131 
132 


so.  in. 
125 


tuns.  ft.  in. 
45  18  140 
16  14*  145 


yr 
54 


rs.  mo.  i».  da. 

11  3  1 
43  11  3  5 


11.  TIME. 


yrs.  days.  li.  min. 
24  352  20  41  20 
14  356  20  49  19 


w.  d.  h.  min.  set. 
472  2  13  18  42 
218  4  16  29  54 


w.  d.  h.  min.  sec. 
781  1  8  23  21 
197  3  12  42  53 


12.    CIRCULAR  MOTION. 

o       o         /         //  S       °  '         " 

9    23    45    54  9    29  3-!-- 

3      7    40    56  7    29  40    36 


QUESTIONS, 

Shewing  the  use  of  Compound  Addition  and  Subtraction. 

NEW-  YORK,  MARCH  22,    1814. 

1.       .  Bought  of  George  Grocer, 

12  C.  2  qrs.  of  Sugar,  at  525.  per  cwt.  :        10 

28  Ibs.  of  Rice,  at  3d.  per  Ib. 
3  loaves  of  Sugar,  wt.  35  Ib.  at  Is.  \d.  per  Ib.   1 
3  C.  2  qrs.  14  Jb.  of  Raisins,  at  36s.  per  cwt.    6       10 


Ans.  41 


0-.  47 

2.  What  sum  added  to  17/.  lls.  8±d.  will  make  100?.  ? 

Ans.  827.  85.  3d.  3qr. 

3.  Borrowed  50/.  10s.  paid   again  at  one  time  17/.  11  s. 
\d.  and  at  another  time,  91.  4s.  'Sd.  at  another  time  171.  9s. 
W   and  at  another  time  19s.  ti±d.  how   much  remains  un- 

id  ?  ^w*.  £4  4s.  9\d. 

4.  Borrowed  100/.  and  paid  in  part  as  follows, viz.  atone 
hne  211.  Us.  Qd.  at  another  time  19/.  17s.  4jd.  at  another 
ime  10  dollars  at  6s.  each,  and  at  another  time  two  English 
guineas  at  28s.  each,  and  two  pistareens,  at  14^6?.  each; 

low  much  remains  due,  or  unpaid  ?     Ans.  £0%  12s.  S^d. 

5.  A,  B,  and  C,  drew  their  prize  money  as  follows,  viz. 
had  75/.  15s.  4(/.     B   had  three  times   as  much  as  A, 

acking  15s.  6d.  and  C,  had  just  as  much  as  A  and  B  both  ; 
iray  how  much  had  C  ?  Ans.  £302  5s.  Wd. 

G.  I  lent  Peter  Trusty   1000  dols.  and  afterwards  lent 
lim  26  dols.  45  cis.  more.     He  has  paid  me  at  one  time 
361  dols.  40  cts.  and  at  another  time  416  dols.  09  cents, 
resides  a  note  which  he  gave  me  upon  James  Paywell,  for 
lols.  90  cts.  ;  how  stands  the  balance  between  us  1 
Ai&.  The  balance  is  $105  06  cts.  due  to  me. 

7.  Paid  A  B,"  in  full  for  E  F's  bill   on  me,  for  105Z.  10s/ 
iz.   I  g.ive  him    Richard  Drawer's  note  for  15J.  14s.  9d. 
?eter  ''Johnson's  do.  for  30/.  Os.  6d.  an  order  on  Robert 
Dealer  for  39/.  Us.  the  rest  I  make  up  in  cash.     I  want  to 

iow  what  sum  will  make  up  the  deficiency  I 

Ans.  £20  3s.  9<2. 

8.  A  merchant  had  six  debtors,  who  together  owed  him 
2917/.  10s.  6d.   A,  B,  C,  D,  and  E,  owed  him  1675 J.  13s. 
9d.  of  ii ;  what  was  F's  debt  ?          Ans.  £1241  16s.  9d. 

9.  A  merchant  bought   17  C.  2  qrs.  14  Ib.  of  sugar,    of 
which  he  sells  9  C.  3  qrs.  25  Ib.  how  much  of  it  remains  un- 

Ans.7C.%qrs.  17  Ib. 

10.  From  a  fashionable  piece  of  cloth  which  contained 

2  na.  a  tailor  was  ordered  to  take  three  suits,  each 
.  2  qrs.  how  much  remains  of  the  piece  ? 

Ans.  32  yds.  2  qrs.  2  na. 

11.  The  war  between  England  and  America  commenced 

0 


48  COMPOUND  I.iULTIPLii:ATIOX. 

April  19,1775,  arid  a  general   peace  took   place  January 
20th,  1783  ;  how  long  did  the  wa^  continue  ? 

Ans.  7  yrs.  9  mo.  I  cL 


COMPOUND  MULTIPLICATION. 
COMPOUND   Multiplication  is  when  the  Multiplicand 
f  consists  of  several  denominations,  &c. 

1 .    To  Multiply  Federal  Money. 

RULE. — Multiply  as  in  whole  numbers,  and  place  the  separatrix  as 
many  figures  from  the  right  hand  in  the  product,  as  it  is  in  the  mul- 
tiplicand, or  given  sum. 

EXAMPLES. 

$  cts.  $   d.  c.  m. 

1.  Multiply  35  09  by  25.         2.  Multiply  49  0  0  5  by  97. 

25  97  ' 


17545  343035 

7018  441045 


Prod.  $877,  25  $4753,  4  8  5 

$.     cts. 

3.  Multiply  1  dol.  4  cts.  by       305         Ans.  317,  20 

4.  Multiply  41  cts.  5  mills  by  150         Ans.     62,  25 

5.  Multiply  9  dollars  by  50         Ans.  450,  00 

6.  Multiply  9  cents  by  50         Ans.       4,  50 

7.  Multiply  9  nulls  by  50         Ans.       0,  45 
8*  There  were  forty-one  men  concerned  in  the  payment 

of  a  sum  of  money,  and  each  paid   3  dollars  and  9  mills  ; 
how  much  was  paid  in  all  1       Ans.  $123  36  cts.  .9  mills. 

9.  The  number  of  inhabitant*  in  the  United  >'.ates  is 
five  millions;  now  suppose  each  should  pay  \  he  trifling 
sum  of  5  cents  a  year,  for  the  term  of  12  years,  towards 
a  continental  tax ;  how  many  dollars  would  b-,>  raised  there- 
by 1  Ans.  Three  millions  Dollars. 

2.  To  Multiply   the  denominations  of  Sterling  Money , 

Weights,  Measures,  fyc. 

RULE. — Write  down  the  Multiplicand,  and  place  the  quantity  un- 
derneath the  least  denominatipn,  for  the  Multiplier,  and  in  multiply- 


COMPOUND  :»Il 'vnPLlCATiOX.  4^ 

ing  by  it,  observe  the  same  rules  for  carrying  from  one  denomination 
to  another,  as  in  compound  Addition.* 

INTRODUCTORY  EXAMPLES. 

£/.  s.    d.  q.  .s.     d. 

Multiply    1  11  6  2  by  5.     How  much  is  3  times      II     S> 


Prod.      £7  17  8  2  £1   15  3 


5.      d.  £.     s>     d.  £.     .<?.      £/. 

15     10     8  24     12     0  -21     15     3 

234 


19     11   10  10     16     4  31     10     9| 

5  6 


31     16     8  1-2     17  10  i4     10  7-i- 

8  9  10 


32     12  10  0     19       1  26     S     4-1- 

11  12  12 


Practical  Questions. 
What  cost  nine  yards  of  cloth  at  5s.  Gd.  per 

£0  5  6  price  of  one  yard, 
Multiply  by  9  yards. 

Ans.  £2  9  6  price  of  nine  yarcta. 

QUESTIONS.  A  \  >  \\  LK >  , 

£.  s.  d.  £.  s.  d. 

4  gallons  of  wine,        at  0     8  7  f>er  gallon.    1   14  4 

5  C.  Malaga  Raisins,  at  1     2  3  percwt.         5  .11  3 
7  reams  of  Paper,         nt  0  17  9^  per  ream.      6    4  6} 

*  When  accounts  are  kept  in  pounds,  shillings,  and  pence,  this  kind  of  mul- 
tiplication is  a  concise  and  elegant  method  oi  finding  thft  value  of  goods,  at 
so  much  per  yard,  Ib.  «fcc.  the  general  rule  being  to  multiply  the  siren  price 
quantity. 

E 


00  CO&flQUXD  MULTIPLICATION,  f 

8yds.  of  broadcloth,  at  1     7     9'   per  yard.  11     24 

9  Ib.  ofj  cinnamon,  at  0  11     4j  per  Ib.  5     2  2} 

11  tuns  jbf  hay,  at  2     1  10     per  tun.  23     0  2 

12  bush/als  of  apples,  at  0     I     9     per  bush.  110 
12  bushels  of  wheat,  at  0    9  10     per  bush.  5  18  0 

2.  When  the  multiplier,  that  is,  the  quantity,  is  a  com- 
posite number,  und  greater  than  12,  take  any  two  such 
numbers  as  v/hen  multiplied  together,  will  exactly  produce 
the  given  quantity,  and  multiply  first  by  one  of  those 
figures,  and  that  product  by  the  other  ;  and  the  last  product, 
will  be  the  answer. 

EXAMPLES. 

What  cost  28  yards  of  cloth,  at  6s.  IQd.  per  yard  ? 

£.  s.  d. 

0  6  10  price  of  one  yard. 
Multiply  by  7 


Produces      2  7  10  price  of  7  yards. 
Multiply  by  4 

Answer,  £$  114  price  of  28  yards 


QUESTIONS. 

ANSWERa. 

5. 

d. 

qrs. 

£. 

s. 

d. 

24 

yards   fit 

7 

4 

3 

per  yard,  = 

S 

17 

6 

27 

—      at 

9 

10 

0 

—        = 

13 

5 

6 

44 

at 

12 

4 

2 

~_        — 

27 

4 

6 

55 

—      at 

8 

3 

1 

—        = 

22 

14 

10J- 

72 

—      at 

19 

11 

0 

—  ,        = 

71 

14 

0 

20 

4-      at 

3 

6 

2 

—        = 

3 

10 

10 

84" 

—     at 

18 

4 

.2 

—        = 

77 

3 

6 

96 

—      at 

11 

9 

0 

—        = 

56 

8 

0 

63; 

—  at£l 

17 

6 

0 

—        =• 

118 

o 

6 

*'44 

—  at     1 

4 

2 

0 

= 

174 

0 

0 

3.  When  no  two  numbers  multiplied  together  will  exactly 
make  the  multiplier,  you  must  multiply  by  any  two  whose 
product  will  come  the  nearest  ;  then  multiply  the  upper 
li»e  by  what  remained  ;  which,  added  to  the  last  product, 
.giyes  the  answer 


vOMl'tHJND  MULTU'I.K    \Tli>\  .') ! 

EXAMPLES. 

What  will  47  yds.  of  cloth  come  to  at  JLf  s.  9df.  per  yd.  fi 
£.     s.      d. 

0  17     9  price  of  1  yatd. 
Multiply  by  5 

Produces      4       89  price  of  5  yards, 
Multiply  by  9 

Produces  39     18    9  price  of  45  yards. 

1  15     6  price  of  2  yards. 

Amvscr,  £41     14     3  price  of  47  yards. 


QUESTIONS. 

ANSWERS 

J 

s. 

d. 

£. 

s. 

d. 

23 

ells 

of  linen., 

at 

0 

3 

6* 

per 

ell. 

4 

1 

5i- 

17 

ells 

of  dowlas, 

at 

0 

1 

6| 

per 

ell. 

1 

6 

oT 

39 

cwt. 

of  sugar, 

at 

3 

19 

6 

per 

cwt. 

137 

9 

(T 

52 

yds. 

of  cloth, 

at 

0 

5 

9 

per 

yd. 

14 

19 

0 

19 

Ibs. 

of  indigo, 

at 

0 

11 

6 

per 

Ib. 

10 

18 

<} 

29 

yds. 

of  cambric, 

at 

0 

13 

7 

per 

yd. 

19 

13 

11 

ill 

yds. 

broadcloth, 

at 

1 

2 

0 

per 

yd. 

124 

17 

<) 

1*4 

beaver  hats, 

at 

1 

9 

4 

u  piece. 

137 

17 

4 

4.  To  find  the  value  of  a  hundred  weight,  by  haying  the 
price  of  one  pound. 

If  the  price  be  farthings,  multiply  2s.  4d.  by  the  farthings 
in  the  price  of  one  Ib. — Or,  if  the  price  be  pence,  multiply 
9s.  4d.  by  the  pence  in  the  price  of  one  Ib.  and  in  (Mtlu^r 
casrj  the  product  will  be  the  answer. 

EXAMPLES. 

1.  What  will  1  cwt.  of  rice  come  to,  at  %\d.  per  Ib.  ? 

s.     d. 

112  farthings=2     4  price  of  1  cwt.  at  ]  d.  per  Ib. 
9  farthings  in  the  price  of  1  Ib. 

Ans.  £1     1     0  price  of  1  cwt.  at  2-Jrf.  per  !b. 


'2  <  '  MULTIPLICATION. 

2.  What  will  1  owl.  of  lead  come  to  at  7d.  per  Ib.  ? 

* .";.  d. 
9  4 


3  5  4 

Questions.  Answers. 

I  cwt.     at  2J-d.  per  Ib.  =  £1     3  4 

I  ditto,   at  2f  d.  —     —  1     5  8 

1  ditto,   at  3d.  —     =  1     b  0 

1    ditto,  at  2d.  —  0  18  8 

1  ditto,  at  3 id.  —     =  1  12  8 

Iltumples  of  Weights,  Measures,  fyc. 
\.   flovv-  much  is  5  times  7  cv/t.  3  qrs.  15  Ib.  ? 
f/#/.  ^-5.  76. 
7     3     15 


]^v.  Cwt.  39     1     19 

/6.  oz.  pwt.  gr.  cwt.  qr.  Ib.  oz. 

:>.  Multijily  <2Q  2     7     13  by  4.  (3)  27  1  13  12 
4  6 


Product  Ib.  80  9  10       4  Ib.  164  0  26     8 


ANSWEf 

yds.  (jr.  na.  yds.  qr.  no. 

4.  Multiply  14     3     2  by  11  'l63     2     2 

lilid.  g.  qt.pt.  lihd.  g.  qt.pt. 

:>.  3Iultiply   21   15  2  1  by  12  254  61  2     0 

/£'.  tn.fur.  po.  le.  m.  fur.  po. 

<>.  Mulriply   81  2     6     21  by  8  655  1     4     8 

a.  i\  p.  a.     r.     p. 

7.  Multiply  41  2  11  by  18  748     0     38 

?//*.    m.  2".    d.  yr.  m.  w.  d. 

8.  Multiply  20     5     3     6  by  14  286     5     2     0 

£.  °      '  >S'.  °    '    '•' 

7  19  2  0 


COMPOUND  DIVISION  63 

cds.  ft.  cds.  ft. 

10.  Multiply  3     87  by  8  29     56 

Practical  Questions  in 
WEIGHTS  AND  MEASURES. 

1.  What  is  the  weight  of  7  lihds.  of  sugar,  each  weigh- 
ing 9  cwt.  3  qrs.  12  Ib.  1  Ans.  69  cwt. 

2.  What  is  the  weight  of  6  chests  of  tea,  each  weighing 
3  cwt.  2  qrs.  9  Ib.  ?  Ans.  21  cwt.  1  qr.  26  Ib. 

3.  How  much   brandy  in   9  casks,   each  containing  41 
gals.  3  qts.  1  pt.  ?  Ans.  376  gals.  3  qts.  Ipt. 

4.  In  35  pieces  of  cloth,  each  measuring  27J  yards,  how 
many  yards  1  Ans.  971  yds.  1  qr. 

5.  In  9  fields,  each  containing  14  acres,  1  rood,  and  25 
poles,  how  many  acres  ?  Ans.  129  a.  2  qrs.  25  rods. 

6.  In  6  parcels  of  wood,  each  containing  5  cords  and  96 
feet,  how  many  cords  ?  Ans.  34 *  cords. 

7.  A  gentleman  is  possessed  of  1 J  dozen  of  silver  spoons, 
each  weighing  2  oz.  15pwt.  11  grs.  2  dozen  of  tea-spoons, 
each  weighing  10  pwt.  14  grs.  and  2  silver  tankards,  each, 
21  oz.  15  pwt.     Pray  what  is  the  weight  of  the  whole  ? 

Ans.  S  Ib.  10  oz.  %pivt.  6  grs. 


COMPOUND  DIVISION, 

TEACHES  to  find  how  often  one  number  is  contained 
in  another  of  different  denominations. 

DIVISION  OF  FEDERAL  MONEV. 

fty  Any  sum  in  Federal  Money  may  be  divided 
whole  number  ;  for,  if  dollars  and  cents  be  written  down  GS 
Q  simple  number,  the  whole  will  be  cents  ;  and  if  the  sum 
consists  of  dollars  only,  annex  two  ciphers  to  the  dollars, 
and  the  whole  will  be  cents  ;  hence  the  following 

GENERAL  RULE. — Writedown  the  given  sum  in  cents,  and  divide 
as  in  whole  numbers  ;  the  quotient  will  be  the  answer  in  cents. 

NOTE.  If  the  cents  in  the  given  sum  are  less  than  10,  you  raiuri 
always  place  a  cipher  on  their  left,  or  in  the  ten's  place  of  the  cen*i 
before  you  write  them  down. 

K  5 


\l)    DIVISION. 
EXAMPLES. 

1.  Divide  35  dollars G8  cents,  by  41. 

41)3568(87     the  quotient  in  cents  ;  and  when  there 

328  is  any  considerable  remainder,  you  may 

annex  a  cipher  to  it,  if  you  please,  and 

288  divide   it  again,  and  you  will  have  the 

287          mills,  &c. 

Hem.        I 

2.  Divide  21  dollars,  5  cents,  by  14. 

14)2105(150     cents— 1  dol.  50  cts.  but  to  bring  cents 
14  into  dollars,  you  need  only  point  off  two 

figures  to  the  right  hand  for  cents,  and 
70  the  rest  will  be  dollars,  &c. 

70 


3.  Divide  4  dols.  9  cts.  or  409  cts.  by  6.      Ans.  68  cts.+ 

4.  Divide  9  dols.  24  cts.  by  12.  Ans.  77  cts. 

5.  Divide  97  dols.  43  cts.  by  85.      Ans.  $1  14  cts.  6m. 
Divide  248  dols.  54  cts.  by  125. 

Ans.  198  cts.  8m.=$l  9S  cts.  8m. 

7.  Divide  24  dols.  65  cts.  by  248;  Ans.  9  cts.  9m. 

8.  Divide  10  dols.  or  1000  cts.  by  25.  Ans.  40  cts. 

9.  Divide  125  dols.  by  50C.  Ans.  25  cts. 
10.  Divide  1  dollar  into  33  equal  parts.         Ans.  3  ctfs.+ 

PRACTICAL    QUESTIONS. 

1.  Bought  25  Ib.  of  coffee  for  5  dollars  ;  what  is  that  a 
pound  ?  Ans.  20  cts. 

2.  If  131  yards  of  Irish  linen  cost  49  dols,  78  cts.  what 
is  that  per  yard  ?  Ans.  38  cts. 

3.  If  a  cwt.  of  sugar  cost  8  dols.  96  cts.  what  is  that  per 
pound  1  Ans.  8  cfs. 

4.  If  HO  reams   of  paper  cost   329  dols.  what   is  that 
per  ream  ?  Ans.  $2  35  cts. 

5.  If  a  reckoning  of  25  dols.  41  cts.  be  paid  equally  among 
14  persons,  what  do  they  pay  apiece?       Ans.  $1  81^  cts. 

6.  If  a  man's  wages  are  235  dols.  80  cts.  a  year,  what  is 
that  a  calendar  month?  Ans.  $19  65  eta. 


7.  The  salary  of  the  President  of  the  United  States,  i.« 
twenty-five  thousand  dollars  a  year ;  what  is  that  a  day  ? 

Ans.  $68  49  cts. 
To  divide  tlie  denominations  of  Sterling  Money, 

Weights,  Measures,  fyc. 

RULE. — Begin  with  the  highest  denomination  as  in  simple  division  ; 
ind  if  any  thing  remains,  find  how  many  of  the  next  lower  denomi- 
nation this  remainder  is  equal  to  ;  which  add  to  the  next  denomina- 
tion :  then  divide  again,  carrying  the  remainder,  if  any,  as  before  ; 
and  so  on  till  the  whole  is  finished. 

PROOF.     The  same  as  in  simple  Division. 

EXAMPLES. 

£  s.     d.     qr.  -    £    s.  d. 

Divide     97  3     11     2  by  5  8)27   18  6 

Quo't.  £19  892                                         £39  9J 

£      s.      d.                                   £     s.  d. 

3.  Divide     31     11     6  by     2             Ans.  15     15  9 

4.  Divide     22       3     9  by     3                        7       7  11 
r>.  Divide     70     10     4  by     4                      17     12  7 

6.  Divide  56  11  5.V  by  5  11  6  3} 

7.  Divide  61  14  8-  by  6  10  5  9[ 

8.  Divide  24  15  6£  by  7  3  10  9J- 

9.  Divide  185  17  6"  by  8  23  4  8} 

10.  Divide  182     16     8  by  9  20       6     3i 

11.  Divide     16       1    11  by  10  1     12     2J 

12.  Divide       1     19     8  by  11  0       3     7£ 

13.  Divide       6       6     6  by  12  0     10     61 

14.  Divide       126  by  9  026: 

15.  Divide  948     11     6  by  12  79       0  11^ 
2.  When  the  divisor  exceeds  12,  and  is  the  product  of  two 

or  more  numbers  in  the  table  multiplied  together. 
RULE. — Divide  by  one  of  those  numbers  first,  and  the  quotient  by 
the  other,  and  the  last  quotient  will  bo  the  answer. 

EXAMPLES. 

£  s.  d.  £   s.  d. 

1.  Divide     29  15  0  by  21  Ans.  1     8   4 

2.  Divide     27  16  0  by  82  0  17   4£ 

3.  Divide     67  9  4  by  44  1108 


£        s.     d.  £      s.     a'. 

4.  Divide     24     16     6     by     36  0     13     91. 

5.  Divide  128       9     0     by     42  31     2* 

6.  Divide  269     12     4     by     56  4     16    3< 

7.  Divide  248     10     8     by     64  3     17     8 

8.  Divide     65     14     0     by     72  0     18     3 

9.  Divide       5     10     3     by     81  O       1     4j 

10.  Divide  115     10     0     by     90  158 

11.  Divide  136     16     6     by  108  154 

12.  Divide  202     13     6     by  121  1     13     6 

13.  Divide     34       4     0     by  144  049 
3.  When  the  divisor  is  large,  and  not  a  composite  num- 
ber, you  may  divide  by  the  whole  divisor  at  once,  after  man- 
ner of  long  division,  as  follows,  viz. 

EXAMPLES. 

1.  Divide  128/.  13s.  3d.  by  47. 

£     s.  d.  £>  s.  d. 
47)128  13  3(2  14  9  quotient 
94 


34     pounds  remaining. 
Multiply  by  20  and  add  in  the  13s. 

produces  (>93 shillings,  which  divided  by  47,  gives- 

47  [14s.  in  the  quotient. 

223" 

188 

35  shillings  remaining. 
Multiply  by  12  and  add  in  the  3d. 

produces  423  pence,  which,  divided  as  abov-e, 

423          [gives  9d.  in  the  quotient. 
£    s.     d.  £  s.     d. 

2.  Divide  113  13     4  by    31  Ans.  3  13     4 

3.  Divide     85    6    3  by     75  129 

4.  Divide  315    3  10J  by  365  0  17    3{ 

5.  Divide  132     0     8  by     68  1  18  10 

6.  Divide  740  16     8  by  100 

7.  Divide  888  18  10  by     95  9    7     ji 


v.O-YlPOUA'D  DIVISION.  57 

Examples    of  Weights,  Measures^  fyc. 

1 .  Divide  1 4  cwt.  1  qr.  8  Ib.  of  sugar  equally  among  8  men 

C.   qr.   Ib.   oz. 

8)14    1     8     0 

1348  Quotient. 

8 
14    1     8     0  Proof. 

2.  Divide  6  T.  11  cwt.  3  qrs.  19  Ib.  by  4. 

Ans.  1  T.    12  cwt.    3  qrs.    25  Ib.  12  oz. 

3.  Divide  14  cwt.   1  qr.  12  Ib.  by  5. 

Ans.  'i  cwt.  3  qrs.  13/6.  9  oz.  9  dr.+ 

4.  Divide  16 Ib.  13 oz.  10 dr.  by6.Ans.2lb.  12 oz.  15 dr. 
6.  Divide  56  Ib.  6  oz.    17  pwt.    of  silver  into  9  equal 

parts.  Ans.  }  Ib.  3  oz.  8  pwt.  13  grs.-f- 

6.  Divide  26  Ib.  1  oz.  5  ,>t.  by  24. 

Ans.  1  Ib.  1  oz.   1  pwt.  1  gr. 

7.  Divide  9  hhds.  28  gals.  2  qts.  by  12. 

Ans.  0  hhd.  49  gals.  2  qts.  1  p/. 

8.  Divide  168  bu.   1  pk.  6  qts.  by  35. 

Ans.  4  bu.  3  pks.  2  <^,?. 

9.  Divide  17  lea.  1  m.  4  fur.  21  po.  by  21. 

Ans.  2  m.  4  fur.  1  jw. 

10.  Divide  43  yds.  1  qr.  1  na.  by  11. 

y^s.  3  yds.  3  tfrs.  3  na. 

11.  Divide  97  E.  E.  4  qrs.   1  na.  by  5. 

Ans.  19  yds.  2  qrs.  3  na.-i- 

J'2.  Divide  4J  gallons  of  brandy  equally  among   14  i 
soldiers.  Ans.  1  gill  aviece. 

13.  Bought  a  dozen  of  silver  spoons,  which  together 
weighed  3  Ib.  2  oz.   13  pwt.   12  grs.  how  much  silver  did 
each  spoon  contain  1  An$,  3  oz.  -ipwt.   11  gr. 

14.  Bought  17  cwt.  C  qrs.  19  Ib.  of  sugar,  and  sold  out 
one  third  of  it ;  how  much  remains  unsold  1 

Ans.  11  cwt.  3  qrs.  2:3  Ib. 

15.  From  a  piece  of  cloth  containing  64  yards  2  na.  a 
tailor  was  ordered  to  make  9  soldiers' coats,  which  took  ono 
third  of  the  whole  piece ;  how  many  yards  did  each  cor»t 

Ans.  2  irrh.    1  nr.  v 


30  -COMPOUND 

PRACTICAL    QUESTIONS. 

1.  If  9  yards  of  cloth  cost  4/.    ,3's.    *i/J.  wliat  is  thai 
per  yard  7 

£      s.     d.     qr. 
9)4     372 

932  Answer. 


2.  If  11  tons  of  hay  cost   23J.  Os.    2</.    what  is  that  per 
tun  ?  Ans.  £2  Is.  lOd. 

3.  If  12  gallons  of  brandy  cost  47.    15s.    6d.    what  is 
that  per  gallon  ?  Ans.  7s.  lid.  %qrs. 

4.  If  841bs.  of  cheese  cost  II.    16s.    Qd.    what  is  that 
>er  pound  ?  Ans.  5}d. 

5.  Bought  48  pairs  of  stockings  for  1 II.  2s.    how  much 
a  pair  do  they  stand  me  in  ?  Ans.  4s.  7^d. 

6.  If  a  reckoning  of  51.  Ss.  W%d.  be  paid  equally  among 
1 3  persons,  what  do  they  pay  apiece  ?        Ans.  Ss.  £±d. 

7.  A  piece  of  cloth  containing  24  yards,  cost    18J.    13s. 
what  did  it  cost  per  yard?  .    Ans.  15s.  3d. 

S.  If  a  hogshead  of  wine  cost  331.   12s.   what  is  it  a  gal- 
lon? Ans.  10s.  Sd. 

9.  If  1  cwt.  of  sugar  cost  3/.   10s.  what  is  it  per  pound 

Ans.  7%d. 

10.  If  a  man  spend  71/.   14s.   6d.  a  year,  what  is  that 
per  calendar  month  ?  Ans.  £5  19s.  Q\d. 

11.  The  Prince  of  Wales'  salary   is  150,000?.   a  year, 
•what  is  that  a  day  ?  AJIS.  £410  19s.  2d. 

12.  A  privateer  takes  a  prize  worth  rxM65  dollars,  of  which 
the  owner  takes  one  half,  the  officers  one  fourth,  and  the  re- 
mainder is  equally  divided  among  the  sailors,  who  ore  125  in 
number ;  how  much  is  each  sailor's  part  1   Ans.  $24  93  cts. 

13.  Three  merchants  A,  B,  and  C,  have  a  ship  in  com- 
pany. A  hath  |,  B  £,  and  C  1,  and  they  receive  for  freight 
228?.  16s.  Sd.      It  is  required  to  divide  it  among  the  own- 
ers according  to  their  respective  shares. 

Ans.  As  share  £143  Os.  5d.      B's  share  £57    4s.    2J, 
f  "*  share  £28  12s.  Id. 

]  I,  A  privateer  haying  taken  a  prize  worth  $6850:,  it 


livided  into  one  hundred  shares  ;  of  which  the  captain  is  to 
lave  11;  2  lieutenants,  each  5;  12  midsipmen,  each  2; 
ind  the  remainder  is  to  be  divided  equally  among  the 
sailors,  who  are  105  in  number. 

Ans.   Captain's  share  $753  50  cts. ;    lieutenant's,  $342 
50  cts.;  a  midshipman's, $137,  and  a  sailor's,  $35  88. 


REDUCTION, 

TEACHES  to  bring  or  change  numbers  from  one  name 
o  another,  without  altering  their  value. 
'  Reduction  is  either  Descending  or  Ascending. 

Descending,  is  when  great  names  are  brought  into  small, 
s  pounds  into  shillings,  days  into  hours,  &c. — This  is  done 
>y  Multiplication. 

Ascending,  is  when  small  names  are  brought  into  great, 

shillings  into  pounds,  hours  into  days,  &c.    This  is  per- 
formed by  Division. 

REDUCTION  DESCENDING. 

RULE. — Multiply  the  highest  denomination  given  by  so  many  of 
tfte  next  less  as  make  one  of  that  greater,  and  thus  continue  till  you 
have  brought  it  down  as  low  as  your  question  requires. 

PROOF.     Change  the  order  of  the  question,  and  divide  your  last 
product  by  the  last  multiplier,  and  so  on. 
EXAMPLES. 

1.  In  25Z.  15s.  9(7.  %qrs.  how  many  farthings! 
£      5.     d.  qrs. 
25     15     9     2  Proof. 

20  4)24758  Ans.  24758. 

515  shillings.            '12)6189  2  qrs. 
12  


6189  pence. 


210)51)5 


-t  £25  15  9  2 

24758  farthings. 
NOTE.  In  multiplying  by  20, 1  added  in  tbe  15s. — by  1% 

the  9d. — and  by  4  the  2qrs.  which  must  always  be  dope  in 
like  csrses. 

In  31Z.  Us.  I0<f.  lgj\  how  many  farthings? 

Ans.  30329. 


60  REDUCT1OA. 

3.  In  46/.  65*  lie?.  Sqrs.  how  many  farthings  I 

Ans.  4444T. 

4.  In  GIL  12s.  how  many  shillings,  pence,  and  farthings  ? 

Ans.  1232s.  14784rf.  59136?™. 

5.  In  84/.  how  many  shilling?-  and  pence  ? 

Ans.  18805.  20160</. 

6.  In  185.  9d.  how  many  pence  and  farthings  1 


7*  In  312Z.  85.  5d.  how  many  half-pence?  Ans.  149962. 

8.  In  846  dollars,  at  6s.  each,  how  many  farthings? 

Ans.  243648. 

9.  In  41  guineas,  at  28s.  each,  how  many  pence? 

Ans.  13776. 

10.  In  59  pistoles,  at  22s.   how  many  shillings,  pence, 
and  farthings  1  Ans.  12985.  15576rf.  62304  qrs. 

11.  In  37  haif-johannes,  at  48s.  how  many  shillings,  six 
pence?,  and  three-pences  ? 

Ans.  17765.  3552  six-pences,  7194  ikrce-pences. 

12.  In  121  French  crowns,  at  65.  8d.  each,  how  man 
pence  and  farthings  ?  Ans.  9380<2.  38720?re. 

REDUCTION  ASCENDING. 

RULE.  —  Divide  the  lowest  denomination  given,  by  so  many  of  tha 
name  as  make  one  of  the  next  higher,  and  so  on  through  all  the  de 
nominations,  as  far  as  your  question  requires. 

PROOF.     Multiply  inversely  by  the  several  divisors. 
EXAMPLES. 

1.  In  224765  farthings,  how  many  pence,  shillings  am 
pounds  ? 

Farthings  in  a  penny  =4)2247- 

Pence  in  a  shilling     —12)5^101   1  qr. 

Shillings  in  a  pound  =2|0)468|2  7d. 

£234  2s.  7d.  I 

Ans.  56191J.  46825.  231.'. 

NOTE.  The  remainder  is.  always  of  the  same  name  r 
the  dividend. 

2.  Brfns  30329  farthings  into  pounds. 

Ans.  £31  11s.  10 


REDUCTION.  61 

8.  In  44447  farthings,  how  many  pounds  ? 

Ans.  £46  55.  lid.  3qrs. 

4.  In   59136   farthings,  how  many  pence,  shillings,  aud 
,ounds?  Aw.l4784rf.  1232s.  £61  12*. 

5.  In  20160  pence,  how  many  shillings  and  pounds  1 

Ans.  1680s.  or  £84. 

6.  In  900  farthings,  how  many  pounds'? 

Ans.  £0  18^.  9(L 

7.  Bring74981  half-pence  iiUopounds..4ns.£l564s.2i</. 

8.  In  243648  farthings,  how  many  dollars  at  6s.  ettcnl 

Ans.  $«46. 

9.  Reduce  13776  pence  to  guineas,  at  28s.  per  guinea. 

Ans.  41. 

10.  In  62304  farthings,  how  many  pistoles,  at  22s.  each? 

Ans.  59. 

11.  In  7104   thive-pences,  how  many  half-johanries,  at 
[8s.  1  Ans.  37. 

•   12.  In  38720  farthings,  how  many  French  crowns,  at 
>s.  8d.  1  Ans.lZl. 

Reduction  Ascending  and  Descending. 

1.    MONEY. 

1.  In  12U  Os,  9id.  how  many  half-pence?  A ns.  58099. 

2.  In  58099  half-pence,  how  many  pounds  ? 

Ans.  121/.  Os.  9.^. 

3.  Bring  23760  half- pence  into  pounds.  Ans.£  19  10^. 

4.  In2l4/;  Is.  3d.  how  many  shillings, six-ponces, three- 
pences, and  farthings'?  Ans.  4281  s.  8562  sir^pc.nces^ 

17125  fhrec-pences,  and  205500  farthings. 

5.  In   1377.    how  many  pence,  and    English   or  French 
crowns,  at  6: 

6.  Is!  249  English  half-crowns,  how  many  pence  and 
pom 

7.  li»   346  guineas,  fit  21  s.  each,  how  many  shillings, 
groats,  and  pence  ?  Ans.TSGGs.  21798  gr'fs    -      '  "7192^ 

8.  In  48  guineas,  at  28s.  '  °ich,  how  many  4^d.  pieces  ? 

Am.  358. 

«  81  guineas,  at  27s.  4d.  each,  how  many  pounds  I 

Ans.  £110  l 


t>2  REDUCTION. 

10.  la  24396  pence,  how  many  shillings,  pounds,  and 
pistoles  1  A?is.  2033^.  £101  13s.  and  92 pistoles.    9s.  over. 

11.  In  252  moidores,  at  36s.  each,  how  many  guineas  at 
28s.  each  1  Ans.  324. 

12.  In  1680  Dutch  guilders,  at  2s.  4d.  each,  how  many 
pistoles  at  22s.  each]  Ans    178 pistoles,  45. 

13.  Borrowed  1248  English  crowns,  at  6s.  8d.  each,  how 
many  pistareens,  at  14|d.  each,  will  pay  the  debt? 

^ws/6885  pistareens,  and  7^d. 

14.  In  50J.  how  many  shillings,  nine-pences,  six-pences, 
four-pences,  and  pence,  and  of  each,  an  equal  number? 

iM.+M.+6d.+4d.+  l<L==9S&.  and  £50= 
12000^.^32=375^5. 

Examples  in  Reduction  of  Federal  Money. 

1.  Reduce  2745  dollars  into  cents. 

2745  dollars  "]      Here  I  multiply  by  100,  the  cents  in 
100  I  a  dollar ;  but  dollars  are  readily  brought 

[into  cents  by  annexing  two  ciphers, 
Ans.  274500  (and  into  mills  by  annexing  three  ci- 

J  phers.  Also,  any  sum  in  Federal  money 
may  be  written  down  as  a  whole  number,  and  expressed  in 
its  lowest  denomination ;  for,  when  dollars  and  cents  aro 
joined  together  as  a  whole  number,  without  a  separatrix, 
they  will  show  how  many  cents  the  given  sum  contains  ; 
and  when  dollars,  cents,  and  mills,  are  so  joined  together, 
they  will  show  the  whole  number  of  mills  in  the  given 
sum. — Hence,  properly  speaking,  there  is  no  reduction  of 
this  money  ;  for  cents  are  readily  turned  into  dollars  by  cut- 
ting off  the  two  right  hand  figures,  and  mills  by  pointing 
off  three  figures  with  a  dot ;  the  figures  to  the  left  hand  of 
the  dot,  are  dollars  ;  and  the  figures  cut  off  are  cents,  or 
cents  and  mills. 

2.  In  345  dollars,  how  many  cents,  and  mills  ? 

Ans.  34500  cts.  345000  mills. 

3.  Reduce  48  dols.  78  cts.  into  cents.  Ans.  4878 

4.  Reduce  25  dols.  8  cts.  into  cents.  Ans.  2508 

5.  Reduce  54  dols.  36  cts.  5m.  into  mills.  Ans.  54365 

6.  Reduce  9  dols.  9  cts.  9m.  into  mills.       Ans.  !*•• 


INDUCTION. 

$  cts. 

7.  Reduce  41925  cents  into  dollars  Ans.  419  25 

8.  Change    4896  cents  into  dollars.  48  96 

9.  Change  45009  cents  into  dollars.  450  09 
10.  Bring       4625  mills  into  dollars.  4  62  5 


2.    TROY    WEIGHT. 

1.  How  many  grains  in  a  silver  tankard,  that  weighs 
1  Ib.   11  oz.  15  pwt. 

Ib.    oz.    pwt. 
I     11     15 
12  ounces  in  a  pound. 

23  ounces. 
20  pennyweights  in  one  ounce 

475  pennyweights. 
24  grains  in  one  pennyweight. 

1900 
950 


Proof.  24)1 1 400  grains.     Ans. 
2,0)47,5 
12)23  15  pwt. 

1  Ib.  11  oz.   15  pwt. 

2.  In  246  oz.  how  many  pwts.  and  grains  1 

Ans.  49ZQpwt.  1 18080 gr* 

3.  Bring  46080  grs.  into  pounds.  Ans.  8. 

4.  In  97397  grains  of  gold,  how  many  pounds  1 

Ans.  16  Ib.  10  oz.  18  pwt.  5  grs. 

5.  In  15  ingots  of  gold,  each  weighing  9  oz.  5  pwt.  how 
many  grains  1  Ans.  66600. 

6.  In  41b.  1  oz.  1  pwt.  of  silver,  how  many  table-spoons, 
weighing  23  pwt.  each,  and  tea-spoons,  4  pwt.  6  grs.  each, 
can  be  made,  and  an  equal  number  of  each  sort  1 

%&pwt.  f  4pwt.  6grs.=654grs.  the  divisor  ;  and  4  Ib.  1  oz. 
1  /^.;=23544£rs,  the  dividend.  Therefore  23544 ~654=- 
of>  Answer 


64  KEDUC/iiuA. 

3.    AVOIRDUPOIS    WEIGHT. 

In  89  cwt.  3'qrs.  14  Ib.  12  oz.  how  many  ounces  ? 
4 

359  quarter*  Proof. 

28  16)161068 

2876  28)10066  12  oz. 

719 


10066  pounds 


4)359  14  Ib. 


16  Ans.  89  cwt.  3  qrs.  14  Ib.  12  oz. 

6039S 
10067  * 

161068  ounces.    Answer. 

2.  In  19  Ib.  14  oz.  11  dr.  how  many  drains?  Ans.  5099. 

3.  In  1  tun,  how  many  drams?  Ans.  573440. 

4.  In  24  tuns.  17  cwt.  3  qrs.   17  Ibs.  5  oz.  how  many 
ounces  ?  .4715.  892245.  * 

5.  Bring 5099  drams  into  pounds.  Ans.  I9lb.  \4toz.  II  dr. 
0.  Bring  573440  drams  into  tuns.  Ans.  1. 

7.  Bring  892245  ounces  into  tuns. 

Ans.  24  tuns,  17  cwt.  3  qrs.  17  Ib.  5oz. 

8.  In  12  hhds.  of  sugar,  each  11  cwt.  25  Ib.  how  many 
pounds  ?  Ans.  15084. 

9.  I:   42  pigs  of  lead,  each  weighing  4  cwt.  3  qrs.  how 
many  foihcr,  at  19  cwt.  2  qrs.  ?    Ans.  10  / 'other,  4£  cwt. 

10.  A.  gentleman  lias  20  hhd^.  of  tobacco,  each  8  cwt. 
3  qrs.   14  Ib.  and  wishes  to  put  it  into  boxes  containing  70 
Ib.  each,  I  demand  the  number  of  boxes  he  must  get? 

Ans.  284. 


4.  APOTHECARIES'  WEIGHT. 

1.  In  9fc  8  3  1  3  2  D  J9  grs.  how  many  grains? 

Ans.  55799. 
£.  In  55799  grains.  ho\v  many  pounds? 

Ans.  9  ft  83  13  29  1 


KEDl  i;!iiK\.  66 

5.    CLOTH    MEASURE. 

1.  In  95  yards,  how  many  quarters  and  nails'? 

Ans.  3SOqrs.  I520w«. 

2.  In  341  yards,  3  qrs.  1  na.  how  many  nails  ? 

Ans.  5460. 

3.  In  3783  nails,  how  many  yard.,  1 

Ans.  236  y ds.  I  qr.  3  na. 

4.  In  61  Ells  English,  how  iminy  (quarters  and  nails  '? 

Ans.  3Q5  qrs.   1220  na. 

5.  In  56  Ells  Flemish,  how  many  quarters  and  nails  ? 

Ans,  168  qrs.  672  na. 

6.  In  148  Ells  English,  how  many  Ells  Flemish  ? 

Ans;  246  E.  F.  2  qrs. 

7.  In    1920  nails,  how  many  yards,  Ells  Flemish,   and 
Ells  English  1      Ans.  120  yds*.  169  E.  F.  and  96  E.  E. 

8.  How  many  coats  can   bo  made  out  of  S6J  yards  of 
broadcloth,  allowing  If  yards  to  a  coat  ?  Ans.  21. 


1.  In  136  bushels,  ho\v  many  pecks,  quarts  and  pints  'I 

Ans.  544pfo.  4352  qts.  87Q4pts. 

2.  In  49 bush.  3pks.  5 qts.  how  many  quarts?  Ans.  1597. 

3.  In  8704  pints,  how  many  bushels  ?  Ans.  136. 
'n  1597  quarts,  how  many  busilota  1 

Ans.  49  bush .  3  pks.  5  qts. 

5.  A  man  would  ship  720  bushels  of  corn  in  barrels, 
which  hold  3  bushels  3  pecks  each,  how  many  barrels 
must  he  get  1  Ans.  192. 

7.  \VINE  MEASURE. 

1.  In  9  tuns  of  wine,  how  many  hogsheads;  gallons  aird 
quarts?  Ans.SGhhds.  2Z68gals.  90720*;. 

2.  In  24  hhds.  18  gals.  2  qts.  how  many  pints  ? 

Ans.  12244. 

3.  In  9072  quarts  how  many  tuns?  Ans.  9. 

4.  In  1906  pints  of  wine,  how  many  hogsheads  ? 

Ans.  3  hhds.  49  gals.  Ipt. 
&*  In  1789  quarts  of  cider,  how  many  barrels? 

Ans.  14  Ms.  25-2*5. 


lib  iiEL»Li/no:v. 

6.  What  number  of  bottles,  containing  a  pint  and  a  halt 
each,  can  be  filled  with  a  barrel  of  cider  ?          Ans.  168. 

7.  Hovv    many  piais,  quarts,   and   two   quarts,  each  an 
equal  number,  may  be  filled  from  a  pipe  of  wine?  Ans.  144. 

8.    LONG    MEASURE. 

1.  Ill  51  miles,  hew  many  furlongs  and  poles? 

Ans.  40H/wr.  10320 poles. 

2.  In  49  yards,  how  many  feet,  inches,  and  barley-corns  ? 

Ana.  U7ft.  1764  inch.-  5292  b.  c. 

3.  How  many  inches  from  Boston  to  New-York,  it  being 
248  miles?  Ans.  15713280  inch. 

4.  In  4352  inches,  how  many  yards  ? 

Ans.  120  yds.  2ft.  8  in. 

5.  In  682  yards,  how  many  rod- 

Ans.  632  x  2 -rl  1=124  rods. 

6.  In  15840  yards,  how  many  miles  and  leagues  ? 

-  Ans.  9  m.  3  lea. 

7.  How  many  times  will  a  carrk.^e  wheel,  Hi  feet  and  9 
inches  in  circumference,  turn   ro^.d  in  going  from  New- 
York  to  Philadelphia;   it  beinjr  9G  tnilcs? 

Ans.  30261  tihics,  and  S\fect  over. 

8.  How  many  barley-corns  will  reach  round   the  globe, 
it  being  360  degrees  ?  Ans.  4755S01600. 

9.  LAND  OR  SQUARE  MEASURE. 

1.  In  241  acres,  3  roods,  and  25  poles,  how  many  square 
rods  or  perches?  Ans.  38705 perches. 

2.  In  20692  square  poles,  how  many  acres  ? 

Ans.  129  a.  1  r.   \2poS. 

3.  If  a  piece  of  land  contain  24  acres,  and  au  enclosure 
of  17  acres,  3  roods,  and  20  rods,  be  taken  out  of  it,  how 
many  perches  are  there  in  the  remainder? 

Ans.  980  perches. 

4.  Three  fields  contain,  the  first  7  acres,  the  second   10 
acres,  the  third   12   acres,   1  rood  ;  how  many  shares  can 
they  be  divided  into,  each  share  to  contain  76  rods  ? 

Ans.  61  shores   and  44  rods  over. 


10.    SOLID    MEASURE. 

1.  In  14  tons  of  hewn  timber,  bow  many  solid  inches  ? 

Ans.  14     50  X  17*28:=:  1 209600. 

2.  In  19  tons  of  round  timber,  how  many  inches'? 

Ans.  1313280. 

3.  In  21  cords  of  wood,  how  many  solid  feet  ? 

Ans.  21     128=2688. 

4.  In  12  cords  of  wood,  how  many  solid  feet  and  inches  ? 

Ans.  153n/i.  and  2054208  inch. 

5.  In  4608  solid  feet  of  wood,  how  many  cords] 

Ans,  36  cds. 

11.  TIME. 

1.  In  41  weeks,  how  many  days,  hours,  minutes,  and 
seconds?  Ans.  287  d.  6888  h.  413280  min.  and  24796800  sec. 

2.  In  214 d.  15 h.  31  m.  25  sec.  how  many  seconds? 

Ans.  18545485  sec. 

3.  In  24796800  seconds,  how  many  weeks?  4>?s.  41  wks. 

4.  In  184009  minutes,  how  many  days? 

Ans.  137 '/.  18  h.  49  min. 

5.  How  many  days  from  the  birth  of  Christ,  to  Christ- 
mas, 1797,  allowing  the  year  to  contain  365 days,  6  hours? 

Ans.  656354  d.  6  7t. 

6.  Suppose  your  age  to  be  16  years   and   20  days,  how 
many  seconds  old  ai-f  you,  allowing  365  days  and  6  hours 
to  the  year?  Aus.  506649600  sec. 

7.  From  March  2d,  to  November  19th  following,  inclu- 
sive, how  many  days  ?  Ans.  262. 

12.  CIRCULAR    MOTION. 

1.  In  7  signs,  15     24'  40",  how  many  degrees,  minutes, 
and  seconds?  Ans.  225°  13524'  and  811480". 

2,  Bring  1020300  seconds  into  signs. 

Ans.  9  signs,  13 '25'. 

Questions  to  exercise  Reduction. 

1.  In  1259  groats,  how  many  farthings,  pence,  shillings. 
and  guineas,  at  28s.  ?      Ans.  20144#rs.  5036<f.  419s.  Sd. 

find  14  guineas,  27s.  8rf. 


>8  UEDUCTiOA 

2.  Borrowed  10  English   guineas  at  28s.  each,  ai)d  24 
English  crowns  at  (is.  and  8d.  each;  how  in  a  jay  p<  stoles  at 
22s.  each,  will  pay  the  debt?  *  Ana.  20. 

3.  Four  UK!.'  brought  each  171. 10s.  sterling  value  in  gold 
into  the  mint,  how  many  « *um< .as  at  2ls.  eaci)  must   tliey 
receive  in  return  t  /jy/s.  (S^  git  in.  14s. 

4.  A   biivcrsinith    received    three    ingots  of  silver,  each 
weighing  27  ounces,   with   directions   to    make   them   into 
spoons  of  2  oz.,  cups  of  5  oz.,  salts  of  1  oz.,  and  snuiF-boxes 
of  2  oz.,  and  deliver  an  equ,-»l  number  of  each  ;  what  was 
the  number?  •/' each,  <mrf  1  oz.  0#er. 

5.  Admit  a  ship's  cargo  from  Bordeaux  to  be  250  pipes, 
130  hhds.  and  150  quarter  cab:.. ,  [  *  i-hds.]  how  many  gal- 
lons in  all ;  allowing  evtry  pint  to  be  a  pound,  what  burden 
ivas  the  ship  of?       Ans.  44415  gals,  and  the  ship's  burden 

158  tons,  12  cwt.  2  qrs. 

G.  In  15  pieces  of  cloth,  each  piece  20  yards,  how  many 
LVencli  Ells  ?  Ans.  200.  " 

7.  In  10  bales  of  cloth,  each   bale   12  pieces,  and  each 

If-mish  Ells,  how  many  yards?         Ans.  2250. 

8.  The  forward  wheels  of  a  wagon  are  141  feet  in   cir- 
romference,  and  the  hind  wheels  15  feet  and  9  inches:  how 
na-iy  more  times  will  the  forward  wheels  turn  round  than 

\  wheels,  in  running  from   ".Boston  to  New- York,  it 

miles  ?  Ans.  7167. 

ow  many  times  will  a  ship  97  feet  6   inches   long, 
•thin  the  distance   of  12800  leagues  and  ten 

Ans.  2079508. 

un  is  95,000,001)  of  miles  from  the  eartfi,  and 

m  ball  at  its  first  d-scbarge  flies  about  a  mile  in  74 

seconds;   bow  long  >v(;u!  i  n  cannon  ball  be,  at  that  rate  in 

;'rom  lit  re  to  the  sun  ?  Ans.  22  yr.  216  d.  12  h.  40 m. 
^  IK  The  sun  .travels   ihrouab  6  signs  of  the   zodiac    in 
tialf  a  Year  ;  bdMmany  decrees,  minutes,  and  seconds  ? 
Aus.  ISO  deg.  10800  min.  648000  sec. 
low  many  strokes  does  a  regular  clock  strike  in  36ft 
lays,  or  a  year  ?  *  Ans.  56940. 

13.  How  long  will  it  take  to  count  a  million,  at  the  rate  of 
50  a  minute  ?          An*.  333  7i.  -20  m.  or  13  d.  21  //.  20  m. 


14.  The  national  debt  of  England  amounts 'to  about  279 
millions  of  pouhds  sterling;  how  long  would  it  take  to  count 
this  debt  in  dollars  (4s.  6d.  sterling)  reckoning  without  in- 
termission twelve  hours  a  day  at  the  i^ite  of  50  dols.  a  mi- 
nute, and  365  days  to  the  year "? 

Ans.  94  years,  134  days,  5 'hours,  20  min. 


FRACTIONS. 

FRACTIONS,  or  broken  numbers,  are  expressions  for 
any  assignable  part  of  a  unit  or  whole  number,  and  (in 
general)  are  of  two  kinds,  viz. 

VULGAR  AND  DECIMAL. 

A  Vulgar  Fraction,  is  represented  by  two  numbers  placed 
one  above  another,  with  a  line  drawn  between  them,  thus, 
J,  f,  &c.  signifies  three  fourths,  five  eighths,  &c. 

The  figure  above  the  line,  is  called  the  numerator,  and 
that  below  it,  the  denominator  ; 

Thus      {  5  Numerator- 

)  8  Denominator. 

The  denominator  (which  is  the  divisor  in  division)  shows 
liow  many  parts  the  integer  is  divided  into ;  and  the  nume.r 
rator  (which  is  the  remainder  after  division)  shows  how  ma- 
ny of  those  parts  are  meant  by  the  fraction. 

A  fraction  is  said  to  be  in  its  least  or  lowest  terms,  when 
t  is  expressed  by  the  least  numbers  possible,  as  £  when  re- 
duced to  its  lowest  terms  will  be  J,  and  Tq¥  is  equal  tof  ,<fec. 

PROBLEM  I. 

To  abbreviate  or  reduce  fractions  to  their  lowest  terms. 
RULE.  —Divide  the  t»;rms  of  the  given  fraction  by  any  number  which 
will  divide  them  without  a  remainder,  and  the  quotients  again  in  the 
same  manner  ;  and  so  on,  till  it  appears  that  there  is  no  number 
greater  than  1,  which  will  divide  them,  and  the  fraction  will  be  in  its 
least  terms. 

EXAMPLES. 
1.  Reduce  4-H  to  its  lowest  terms. 

-  (3)   (2) 

8)4H=^f=T6o-f  the  Answer. 
:4.  Reduce* iff  to  its  lowest  terms.  An* 

3.  Reduce  \\%  to  its  lowest  terms.  Ans.     J 

its  lowest,  terms, 


5.  Abbreviate  ff  as  much  as  possible.  An*. 

C.  Reduce  Jff  to  its  lowest  terms.  Ans.  | 

7.  Reduce  Jf  f  to  its  lowest  terras.  Ans. 

8.  Reduce  £f*  to  its  lowest  terms. 

9.  Reduce  -ff  £  to  its  lowest  terms.  . 
10.  Reduce  £f  f  J  to  its  lowest  terms.  Ans. 

PROBLEM  II. 

To  find  the  value  of  a  fraction  in  the  known  parts  of  the 
integer,  as  to  coin,  weight,  measure,  &c. 

RULE. — Multiply  the  numerator  by  the  common  parts  of  the  integer, 
and  divide  by  the  denominator,  £c. 

EXAMPLES. 

1.  What  is  the  value  of  4  of  ,?  pound  sterling? 
Numer.       2 

20  shillings  in  a  pound. 

Denoin.  3)IO(13s.4d.  Ans. 
3 

10 
9 


3)12(4 
12 

2.  What  is  the  value  of  f  ^  of  a  pound  sterling 

Ans.  ISs.  5d.  2T\  qrs. 

3.  Reduce  %  of  a  shilling  to  its  proper  quantity.  Ans.  Qd. 

4.  What  is  the  value  of  f  of  a  shilling'?  Ans.  4±d. 

5.  What  is  the  value  of  ||  of  a  pound  troy?  Ans.  Qoz. 

6.  How  much  is  T9T  of  a  hundred  weight? 

Ans.  3  qrs.  7  Ib.  10^T  oz. 

7.  What  is  the  value  of  £  of  e  mile  ? 

.0/wr.  26  po.  11/f. 

8.  How  much  is  |  of  a  cwt.  ?  Ans.  3  <//\<?.  3  /6. 1  oz.  12|  </r. 

9.  Reduce  f  of  an  Ell  English  to  its  proper  quantity. 

Ans.  2  qrs.  3±na. 
10.  How  much  is  4  of  a  hhd.  of  wine  ?          Ans,  54  gat. 


</ 


11.  What  is  the  value  of  -^  of  a  day  I 

Ans.  16  h.  36  min.  55j'\j  sec. 

PROBLEM  in. 

To  reduce  arty   given  quantity  to  the  fraction   of  any 
Teater  denomination  of  the  same  kind. 

RULE.  —  Reduce  the  given  quantity  to  the  lowest  term  mentioned 
or  a  numerator  ;  then  reduce  the  integral  part  to  tae  same  term,  for  a 
lenominator  ;  which  will  be  the  fraction  required. 
EXAMPLES. 

1.  Reduce  13s.  6d.  2qrs.  to  the  fraction  of  a  pound. 
20  integral  part  -  13  6  2  given  sum. 
12  12 

240  162 

4  4 


960  Denominator.  650  Num.       Ans.  f  £  f= 

2.  What  part  of  a  hundred  weight  is  3  qrs.  14  Ib.  ? 

3  qrs.  } 4 lb.— 98  Ib.         Ant.  T9T82=f 

3.  What  part  of  a  yard  is  3  qrs.  3  na.  1  Ans.  || 

4.  What  part  of  a  pound  sterling  is  13s.  4d.  ?     Ans.  f 

5.  What  part  of  a  civil  year  is  3  weeks,  4  days  ? 

Ans.  ^=^3- 

6.  What  part  of  a  mile  is  6  fur.  26  po.  3  yds.  2  ft.  ? 
fur.  po.  yds.  ft.     feet. 

(>     26       3      2=4400  Num. 

a  mile  =5280  Denom.  Ans.  fJfS=| 

7.  Reduce  7  oz.  4  pvvt.  to  the  fraction  of  a  pound  troy. 

Ans.  J 

8.  What  part  of  an  acre  is  2  roods,  20  poles  ]     Ans.  | 

9.  Reduce  54  gallons  to  the  fractioivof  a  hogshead  of 
vine. 

10.  What  part  of  a  hogshead  ;s  9  gallons  ?  Ans.  \ 

11.  What  part  of  a  pound  troy  is  10  oz.  10  pwt.  10  grs, 

Ans. 

DECIMAL  FRACTIONS. 

\DecimalFraction  is  that  whose  denominator  is  a  unit, 
ith  a  cipher,  or  ciphers  annexed  to  it.  Thus, 


/2  FRACTIONS. 

The  integer  is  always  divided  either  into  10,  100,  1000, 
&c.  equal  parts;  consequently  the  denominator  of  the  frac- 
tion will  always  be  either  10,  100,  1000,  or  10000,&c.  which 
being  understood,  need  not  be  expressed ;  for  the  true  value 
of  the  fraction  may  be  expressed  by  writing  the  numerator 
only  with  a  point,  before  it  on  the  left  hand  thus,  T\  is  writ- 
ten ,5  ;^  TVV  ,45  ;  -^Tt  ,725,  <fcc. 

But  if  the  numerator  has  not  so  many  places  as  the  de- 
nominator has  ciphers,  put  so  many  ciphers  before  it,  viz. 
at  the  left  hand,  as  will  make  up  the  defect;  so  write  Tf7 
thus,  ,05;  and  -^T7r  thus,  ,0^6,  &c. 

NOTE.     The  point  prefixed  is  called  the  separatrix. 

Decimals  an  counted  from  the  left  towards  the  right 
hand,  and  t-aeii  figure  takes  its  value  by  its  distance  from 
the  unit's  place  ;  if  it  he  i.i  tl»vj  first  place  after  ur,.tN  (or  se- 
parating point)  it  sigmfifr;  tenths;  if  in  the  scroi.il,  hun- 
dretlihs,  &c.  drrrenaing  in  each  place  in  a  tenfold  propor- 
tion, as  in  the  following 

NUMERATION  TABLE. 

'n    en 

t:  tS 


zr  &  s  -^  ~E a 

7  654321  234567 

Whole  numbers.  Decimals. 

Ciphers  placed  at  the  right  hand  of  a  decimal  fraction 
do  not  alter  its  value,  since  every  ^>g;;ificar,t  figure  conti- 
nues to  possess  the  same  place  :  so  ,5  ,50  and  ,500  are  all 
the  same  value,  n-u!  equal  to  /„  or  J. 

But  ciphers  placed  at  the  left  hand  of  decimals,  decrease 
their  value  ir  a  tenfold  proportion,  by  ren.ov  ng  them  fur- 
ther from  the  decimal  point.  Thus,  ,5  ,05  ,005,  &e.  are 
five  tenth  parts,  five  hundredth  parts,  five  thousandth  parts, 
<fec.  respectively.  It  is  therefore  evident  that  the  magnitude 


TJ  E  v,  1  ;»1  A  L   K  li A C  XI  i ;  73 

<tf  a  dcuunal  fraction,  compared  with  another,  Joqs  not  de- 
pend upon  the  number  of  its  figures,  but  upon  the  value  of 
its  first  left  hand  figure  :  for  instance,  a  fraction  beginning 
•sv ill \&uy  figure  less  than  ,9  such  as  ,899229,  &c.  if  extend- 
ed to  an  infinite  number  of  figures,  will  not  equal  ,9. 

ADDITION  OF  DECIMALS. 

RULE. — 1.  Place  the  numbers,  whether  mixed  or  pure  decimals,  un- 
der each  other,  according  to  the  value  of  their  places. 

2.  Find  their  sum  as  in  whole  numbers,  and  point  off  so  many  places 
for  the  decimals,  as  are  equal  to  the  greatest  number  of  decimal  patts 
in  any  of  the  given  numbers. 

EXAMPLES 

1.  Find  the  sum  of  41,653 +36 ,054-24,000+1, 6 
f  41,653 
j  38,05 
i  24,009 
I   1,6 


&um,  103,312,  which  is  103  integers,  and  TV#o-  p'ai'ts  of 
a  unit.  Or,  it  is  103  units,  and  3  tenth  parts,  1  hundredth 
phrt,  and  2  thousandth  parts  of  a  unit,  or  1. 

Hence  we  may  observe,  that  decimals,  and  FEDERAL 
MONEY,  are  subject  to  one  and  the  same  law  of  notation, 
and  consequently  of  operation. 

For  since  dollar  is  the  money  unit ;  and  a  dime  being  the 
tenth,  a  cent  the  hundredth,  and  a  mill  the  thousandth  part 
of  a  dollar,  or  unit,  it  is  evident  that  any  number  of  dollars, 
dimes,  cents  and  mills,  is  simply  the  expression  of  dollars, 
and  decimal  parts  of  a  dollar  :  Thus,  11  dollars,  6  dimes., 
5  cents,=ll,65  or  11TW  dol.  &c. 

2.  Add  the  following  mixed  numbers  together. 

(2)  (3)  (4) 

Yards.  Ounces.  Dollars. 

46,23456  12,3456  48,9108 

24,90400  7,891  1,8191 

17,00411  2,34  3,1030 

3,01111  5,6  ,7012 


5.  Add  the  following  sums  of  Dollars  together,  viz, 
$12,34565+7,891  +2,34+ 14,  +  ,001 1 

Ans.  $36,57775,  or  $36,  5di.  lets.  l^mliU. 

6.  Add  the  following  parts  of  an  acre  together,  viz. 

,7569  -H25+ ,654+, 199.       Ans1\  ,8599  acres.. 

7.  Add  72,5+3^714-2,15744- 371,4+2,75. 

7  Ans.  480,8784 

.   8.  Add  30,07^00,71+59,4+3207,1.     Ans.  3497,28 
9.Add71,467+27,94+16,084+98,009+86,5.,l7,'s.300 

10.  Add  ,7509+, 0074+, 69+, 8408+, 6109.    Ans.Z$ 

11.  Add  ,6+,099+,37  +  ,905+,Q26.  4ns.  2 

12.  To  9,999999  add  one  millionth  part  of  a  unit,  and 
ilie  sum  will  be  10: 

13.  Find  the  sum  of 

Twenty-five  hundredths,     ---„-- 
Three  hundred  and  sixty-five  thousandths, 
Six  tenths,  and  nine  millionth;?.    -     -     -     - 

Ans.  1,215009 
SUBTRACTION  OF  DECIMALS. 

HULE.' — Place  the  numbers  according  to  their  value  ;  tluTi  su 
,  whole  numbers,  and  point  off  the  decimals  as  in  Add;;. 
EXAMPLES. 

Dollars.  IncJics. 

1.  From    125,64  2.  From  14,674 

Take     95,58756  Take    5,91 


3.  From   761,8109  719,10009  27,15 

Take      18,9113  7,121  1,51679 


0.  From  480  take  245,0075  Ans.  234,9925 

7.  From  236  dols.  take  ,549dols.  Ans.  $235,451 

8.  From  ,145  take  ,09684  Ans.  ,04816 

9.  From  ,2754  take  ,2371  Ans.  ,0383 

10.  From  271  take  215,7  Ans.  55,3 

11.  From  270,2  take  75,4075  Ans.  194,7925 
12   From  107  take  .0007  Ans.  106.9993 


7a 

13.  From  a  unit,  or  1.  subtract  the  millionth  part  of  it- 
self. AKS.  ,999099 

MULTIPLICATION  cfip  DECIMALS. 

HULE. — 1.  Whether  they  bo  mixed  numbers,  or  pure  decimals,  place 
the  factors  and  multiply  them  as  in  whole  numbers. 

2.  Point  oil  so  many  figures  from  the  product  as  there  are  decimal 
places  in  both  the  factors;  and  if  there  be  not  so  many  places  in  the 
product,  supply  the  defect  by  prefixing  ciphers  to  the  left  hand. 
EXAMPLES. 

1.  Multiply     5,230  5.  Multiply     3,024 

b>       ,098  by       2,23 


Product,  ,041883  6,74352 

3.  Multiply  25,238  by  12,1.7.          Ayisieers.  307,14646 

4.  MiiUiniy  2461  by  ,0529.  130,1869 

5.  Multiply  7853  by  3.5.  27485,6 
i).  Multiply  ,007853  by  ,0:1.5.                        ,000274855 
7.  Multiply  004  by  ,004.  ,000016 

•/hut*  cost  6,21  yards  of  cloth,  at  2  dols.  32  cents,  5 
iijs,  per  vard  ?  Ans.  §14,  4d.  8e.  S^\m. 

1>.  j^ultiply  7,0;>  dollars  by  5.27  dollar?. 

Ans.  35,9954  dob.  or  $36  9D  ets.  5, 
li).  Multiply  41dols.  25cts.  by  120  dollars.  Ans.  §4950 

11.  Multiply  3  dols.  45  els.  by  16  cts. 

yl/w,  "$0,5520=55  els. 

12.  Multiply  65  cents,  by  ,09  or  9  cents. 

Ans.  $0,0585—5  cts.  $±  mills. 

13.  Multiply  10  dels,  by  10  cts.  Ans.  $1 
1 4:"ktultiply  341 ,45  dols.  by  ,007  or  7  mills.  Ans.  $2,39 
To  multiply  by  10, 100,  1000,  &c.  remove  the  separating. 

point  so  many  places  to  the  right  hand,  as  the  multiplier 
has  ciphers. 

(  Multiplied  by  10,       makes  4.25 

So  ,425  I   — by  100,     makes  42,5 

( by  1000,  is  ,425 

For     ,425X10  Is  4,250,  &o. 

DIVISION  OF  DECIM  \ 

llui.E. — 1.  The  places  of  the  decimal  parts  oi 


JJ.KJJIMA.L  FRACTIONS. 


divide  as  in  whole  numbers,  and  from  the  right  hand  of  the 
quotient,  point  off  so  many  places  for  decimals,  as  the  decimal  places 
ill  the  dividend  exceed  those  in  the  divisor. 

2.  If  the  places  in  the  quotient  be  not  so  many  as  the  rule  requires, 
supply  the  defect  by  prefixing  ciphers  to  the  left  hand  of  said  quotient. 

NOTE. — If  the  decimal  places  in  the  divisor  be  more  than 
Chose  in  the  dividend,  annex  as  many  ciphers  to  the  divi- 
dend as  you  please,  so  as  to  make  it  equal,  (at  least,)  to  the 
divisor.  Or,  if  there  be  a  remainder,  you  may  annex  ciphers 
-to  i,t,  and  carry  on  the  quotient  to  any  degree  of  exactness. 


EXAMPLE-. 


9,51)77,4114(3.11 

7(3,08 


3.8),21316(,0561 
190 


1,331 
951 

3804 
3804 


328 

38 
38 


00 

Answers.     32,12 
,23068+ 
,00758 
,00150+ 
,038356  + 
,40736+ 
611,9+ 


3.  Divide  780,517  by  24,3. 

4.  Divide  4,1  8  by  ,1812. 
3.  Divide  7,25406  by  957. 

(>.  Divide  ,00078759  by  ,525. 

7.  Divide  14  by  365. 

8.  Divide  $246,1476  by  $604,25. 

9.  Divide  $186513,239  by  $304,81. 
U>.  Divide  $1,28  by  $8,31 

11.  Divide  56  cts.  by  1  dol.  12  cts. 

12.  Divide  1  dollar  by  12  cents. 

13.  If  213  or  21,75  yajds  of  cloth  cost  34,317  dollars, 
what  will  one  yard  cost  1  $  1  ,577  + 

NOTE.-—  When  decimals,  or  whole  numbers,  are  to  be  di- 
^ided  by  10,  100,  1000,  &e.  (viz.  unity  witli  ciphers,)  it  i* 
performed  by  removing  the  separatrix  in  the  dividend,  sc 
many  places  towards  the  left  hand  as  there  one  ciphers  in 
the  divisor. 


8,333  + 


r  10,     the  quotient,  is  57, ~ 

57-2  divided  by    J  100,      -     -     -     -      5,7:2 

f  1000,    -     -     -     -      ,57:2 


REDUCTION  OF  DECIMALS. 

CASE    I. 

To  reduce  a  Vulgar  Fraction  to  its  equivalent  Decimal. 

RULE. — Annex  ciphers  to  the  numerator,  and  divide  by  the  deno- 
ninator;  and  the  quotient  will  be  the  decimal  required. 

NOTE. — So  many  ciphers  as  you  annex  to  the  given  ny- 
ucrator,  so  many  places  must  be  pointed  in  the  quotient : 
irul  if  there  be  not  so  many  places  of  figures  in  the  quotient, 
nakc  up  the  deficiency  by  placing  ciphers  to  the  left  hand 
.if  the  said  quotient. 

EXAMPLES. 

1.  Reduce  -J-  to  a  decimal.  8)1,000 

Ans.  ,1.25 
^.  What  decimal  is  equal  to  J  ?  Ansivers.     ,5 

3.  What  decimal  is  equal  to  J  ? ,75 

4.  Reduce  4-  to  a  decimal.          ------       .2 

5.  Reduce  -j -J-  to  a  decimal.        -----       />£75 

6.  Reduce  ~-J  to  a  decimal.        ---....,     5g£> 

7.  Bring  •£%  to  a  decimal.  -----    ,00375 

8.  What  decimal  is  equal  to  ^T  1    -     -     -    ,0370*37+ 
1).  Reduce  ^  to  a  decimal.         -  '  *•     -     -    ,333333 -j- 

10.  Reduce  T^J  to  ^s  equivalent  decimal.     -     -  ,008 

11.  Reduce  2*Vto  a  decimal.        -     -     -       .  1923076 -f- 

CASE  II. 

To  reduce  quantities  of  several  denominations  to  a  Decimal* 
RULE. — 1.  Bring  the  given  denominations  first  to  a  vulgar  fraction 
by  Problem  HI.  page  71 ;  and  reduce  said  vulgar  fraction  to  its  equi- 
valent decimal ;  or, 

2.    Place  tfto  several  denominations  above  each  other,  lotting  U;e 
highest  denomination  stand  at  the  bottom  ;  then  divide  each  denomi- 
nation (beginning  at  the  top)  by  its  value  in  the  next  dcaoinh' 
the  last  ouotie'nt,  vrjll  give  the  decimal  reoui;- 


3DECIAI AL  jbUiACiTO.N S . 


EXAMPLES. 

1.  Keduce  1'2  §.  6d.  '3qrs.  to  the  decimal  of  a  pound. 


150 
4 


5760 

2700 
1920 

7800 
768.0 


Answer. 


By  Rule 
4    3, 


*400 
1920 

4800 
4800 


2.  Ileduce  los.  9d.  3qrs.  to  the  decimal  of  a  pound. 

Ans.  ,700625 

3.  Reduce  Dd.  3  qrs.  to  the  decimal  of  a  shilling. 

-4ns.  ,8125 

4.  Jleduce  TJ  farthings  to  the  decimal  of  a  shilling. 

Ans.  ,0625 

5.  Reduce  3s.  4d.  New-England  currency,  to  the  deci- 
mal of  a  dollar.  Ans.  ,555555+ 

6.  Reduce  12s.  to  the  decimal  of  a  pound.         Ans.  ,6 
NOTE. — When  the  shillings  are  even,  half  the  number, 

with  a  point  prefixed,  is  their  decimal  expression  ;  but  if  the 
number  be  odd,  annex  a  cipher  to  the  shillings,  and  then 
by  halving*  them,  you  will  have  their  decimal  expression-. 

7.  Reduce  1,  2,  4,  9, 16  and  19  shillings  to  decimals. 
Shillings        1         2        4        9         16         19 

,'0.5    ,1    .2   ,45   ,8    ,95 


Dfc£:JMAL   FRACTIONS.  <J 

5.  \Vhat  is  the  decimal  expression  of  4?.  19s.  6i<L  ? 

4ns.  £4,97708+ 
9.  Brin"  34/.  16s.  7fd.  into  a  decimal  expression. 

Ans.  £34,8322916+ 

10.  Reduce  25/.  19s.  5^d.  to  a  decimal. 

Ans.  £25,972916+ 

1 1.  Reduce  3  qrs.  2  ria.  to  the  decimal  of  a  yard.  Ans.  ,875 
1&  Reduce  1  gallon  to  the  decimal  of  a  hogshead. 

Ans.  ,015873+ 

13.  Reduce  7  oz.  19  pwt.  to  the  decimal  of  a  Ib.  troy. 

Ans.  ,6625 

14.  Reduce  3  errs.  21  Ib.  avoirdupois,  to  the  decimal  of  a  cwt. 

Ans.  ,9375 

15.  Reduce  2  roods,  16  perches,  to  the  decimal  of  an  acre, 

Ans.  ,6 
ftf.  Reduce  2  feet  6  inches  to  the  decimal  of  a  yard. 

Ans.  ,833333+ 

17.  Reduce  5  fur.  16  po.  to  the  decimal  of  a  mile.  Ans.  ,675 

18.  Redirce  4*  calendar  months  to  the  decimal  of  a  year. 

Ans.  ,375 


CASE  III. 

r£o£nd  the  value  of  a  Decimal  in  the  knownparts  eftlie  In- 
teger. 

RULE. — 1.  Multiply  the  decimal  by  the  number  of  parts  in  the  next 
Jes*  denomination,  and  cut  off  so  many  places  for  a  remainder,  to  tho 
rigjit  hand,  as  there  are  places  in  the  given  decimal. 

2.  Multiply  the  remainder  by  the  next  inferior  denomination,  and 
cut  off  a  remainder  as  before  ;  and  so  on  through  all  the  parts  of  the 
integer,  and  the  several  denominations  standing  on  the  left  hand 
make  the  answer. 

EXAMPLES. 

V  What  is  the  value  of  ,5724  of  a  pound  sterling  ? 
£  ,5724 
20 


11,4480 
12 

5-,37T6*0  [Carried  up.] 


30  DECIMAL 

5,3760 


1,5040  Am.  Us.  5d.  l,50r*. 

2.  What  is  the  value  of  ,75  of  a  pound  ?        Ans.  15s. 

3.  What  is  the  value  of  ,85251  of  a  pound  ? 

Ans.  17  s.  Oa'.  2,4  <?r*. 

4.  What  is  the  value  of  ,040625  of  a  pound  1     Ans.  9$d. 

5.  Find  the  value  of  ,8125  of  a  shilling.         Ans.  9*d. 

6.  What  is  the  value  of  ,617  of  a  cwt.  ? 

Ans.  2qrs.  13  Ib.  I  oz.  10,6  dr. 

7.  Find  the  value  of  ,76442  of  a  pound  troy. 

Ans.  9  oz.  opwt.  llgr-. 

8.  What  is  the  value  of  ,875  of  a  yd.  ?  Ans.  3  qrs.  2  ntt. 

9.  What  is  the  value  of  ,875  of  a  hhd.  of  wine? 

Ans.  56  gals.  Qqt.  Ipt*. 

10.  Find  the  proper  quantity  of  ,089  of  a  mile. 

An  si  28  po.  2yds.  I  ft.  ll,04i//. 

11.  Find  the  proper  quantity  of  ,9075  of  an  acre. 

Ans.  3r.  25,2po. 

12.  What  is  the  value  of  ,569  of  a  year  of  365  days! 

Ans.  207  d.  16  h.  26  m.  24  sec. 

13.  What  is  the  proper  quantity  of  ,002084  of  a  pound  troy? 

Ans.  12,00384  gr.~ 

14.  What  is  the  value  of  ,046875  of  a  pound  avoirdupois? 

Ans.  12  dr. 

15.  What  is  the  value  of  ,712  of  a  furlong? 

Ans.  2Spo.  2yds.  I  ft.  ll,04/». 

16.  What  is  the  proper  quantity  of  ,142465  of  a  year  ? 


CONTRACTIONS  IN  DECIMALS. 

PROBLEM    I. 

A  CONCISE  and  easy  method  to  find  the  decimal  of  any 
number  of  shillings,  pence  and  farthings,  (to  three  places) 
by  INSPECTION. 

RULE.  —  1.  Write  half  the  greatest  even  number  of  shillings  for  £ho 
first  decimal  figure. 

2.  Let  the  farthings  in  the  given  pence  and  farthings  possess  the 
second  a?id  third  places;  observing  to  increase  the  sreccrrrtl  pla-ee  or 


DECIMAL  FRACTION.:-..  81 

place  of  hundredths,  by  5,  if  the  shillings  be  odd  ;  and  the  third  place 
by  1  when  the  farthings  exceed  12,  and  by  2  when  they  exceed  36. 
EXAMPLES. 

1.  Find  the  decimal  of  7s.  9jd.  by  inspection. 

,3     =1-    6s. 

5       for  the  odd  shillings. 

39=the  farthings  in  9jd. 

2     for  the  excess  of  30. 

£.  ,391  r=r decimal  required. 

2.  Find  the  decimal  expression  of  IGs.  4^d.  and  17s.  8^ 

-4ns.  £  ,819,  and  £  ,885 
t>.  Write  down  £47  18  10]  in  a  decimal  expression. 

Ans.  £47,943 
>.  Reduce  £\  8s.  2d.  to  an  equivalent  decimal. 

Ans.  £1  ,40 


PROBLEM  II. 

.1  short  and  easy  method  to  find  the  value  of  any  decimal  of 

a  pound  by  inspection. 

RULE. — Double  the  first  figure,  or  place  of  tenths,  for  shillings,  and 
if  the  second  figure  be  5,  or  more  than  5,  reckon  another  shilling ;  then, 
after  this  5  is  deducted,  call  the  figures  in  the  second  and  third  places 
«o  many  farthings,  abating  1  when  they  are  above  12,  and  2  when 
above  36,  and  the  result  will  be  the  answer. 

NOTE. — When  the  decimal  has  but  2  figures,  if  any  thing 
remains  after  the  shillings  are  taken  out,  a  cipher  must  be 
annexed  to  the  left  hand,  or  supposed  to  be  so. 

EXAMPLES. 

1.  Find  the  value  of  £.  ,679  by  inspection. 
12s=double  of  6 
.    1         for  the  5  in  the  second  place  which  is  to  be 

[deducted  out  of  7 

Add         7Jd.— 29  farthings  remain  to  be  added. 
Deduct      Jd.  for  the  excess  of  12. 


Ans.  13s.  ~d. 

'2.  Find  the  value  of  £.  ,870  by  inspection.  Ans.  17s.Gjf7. 
IK  Find  the  value  of  £.  ,842  by  inspection.  Ans.  16$.  Wd. 
V  Find  the  value  of  £.  ,.097 by  inspect! on.  Aits, }?.  11  W. 


REDUCTION  OF  CURRENCIES. 

RULES  for  reducing  the  Currencies  of  the  several  United 
States*  into  Federal  Money. 

CASE  I. 

To  reduce  the  currencies  of  the  different  states,  where  a 
crollaris  an  even  number  of  shillings,  to  Federal  Money. 
They  are 

f  New-England,  Nero-  York,  and  ) 

J  Virginia,  North  Carolina.  ( 

*j  Kentucky,  and 
£  Tennessee. 

RULE. — 1.  When  the  sum  consists  of  pounds  only,  annex  a  ciphn 
to  the  pounds,  and  divide  by  half  the  number  of  shillings  in  a  dollar; 
the  quotient  will  bo  dollars. t 

^  2.  But  if  tiie  eiim  consists  of  pounds,  shillings,  pence,  &c.  bring  the 
given  sum  into  shillings,  and  reduce  the  pence  and  farthings  to  a  de- 
cimal of  a  shilling  ;  annex  said  decimal  to  the  shillings,  with  a  decimal 
point  between,  then  divide  the  whole  by  the  number  of  shillings  con- 
tained in  a  dollar,  and  the  quotient  will  be  dollars,  cents,  mills.  &:c. 
EXAMPLES. 

1.  Reduce  737.  New-England  and  Virginia  currency,  to 
Federal  money.  3)730 

|       cts. 

$243j-243  83 1 

2.  Reduce  45/.  155. 1\d.  New-England  currency,  to  fedc- 

20  [ral  monev. 

d. 

A  dollar=6)91 5,625  12)7,500 

$152,6044-     Ans.  ,625  decimal. 

*  Formerly  the  pound  was  of  the  same  sterling  value  in  all  the  colonies 
as  in  Great-Britain,  and  a  Spanish.  Dollar  worth  4s.  6d. — but  the  legisla- 
tures of  the  different  colonies  emitted  bills  of  credit,  which  afterwards  de- 
preciated in  their  value,  in  some  states  more,  in  others  less,  &£. 
Thus  a  dollar  is  reckoned  in 


Jfeic-England. 
Virginia, 
Kentucky,  and 
Tennessee, 


New-Jersey, 
Pennsylvania, 
Delaware,  and 
Jilaryland, 


South- 


Georgia, 


^few-York,  and     fi 

jforth  Carolina, )  Cl 

t  Adding  a  cipher  to  the  pounds,  multiplies  the  whole  ny  10, 
them  into  tenths  of  a  pound ;  then  because  a  dollar  is  just  three  tenth?  of:' 
pound,  N.  Li.  currency,  dividing1  throse  ifntbs  byS?brih^the!fl  inio(Muar.« 


1  farthing  is  ,25  J  which  annex  to  the  pence,  and 

2  —       —  ,50  >  divide  by  12,  you  will  have  the 

3  —       =  ,75  j  decimal  required. 

3.  Reduce  345/.  10s.  ll\d.  New-Hampshire,  &c.  curren- 
^v  to  Spanish  milled  dollars,  or  federal  monev. 
£345  10  III 

20  cL 

12)11,2500 


6)6910,937^ 


,9375  decimal 


$1151,8229+  Ans. 

4.  Reduce  105Z.  14s.  3*d.  New-York  and  Nortb-Caroli- 
la  currency,  to  federal  money. 

,£105     14     3J-  d 

20  12)3,7500 

A  doliar=8)21 14,3125  ,3125  decimal 

$264,28906  Ans. 
Or  $  dcm.  TW 

5.  Reduce  4317.  New-York  currency  to  federal  money. 
rhis  being'  pounds  only,* —       4)4310 


Ans. 

C.  Reduce  2SZ.  lls.  6d.  New-England  and  Virginia  cur- 
ency,  to  federal  money.  Ans.  $95,  25  cts. 

7.  Change  463/f.  10^.  Sd.  New-England,  &c.  currency, 
o  federal  money.  Ans.  $1515,  llefa.  lm.-r 

8.  Reduce  35/.   19s.  Virginia,  &c.  currency,  to  federal 
noncy.  -4ns.  $119,  83  cts.  3  m.  -f 

9.  Reduce  214?.  10s.  7±d.  New- York,  &c,  currency,  to 
edoral  money.  Ans.  $536,  32  cts.  8  w^.+ 

10.  Reduce  304?.  11s.  5tf.  North-Carolina,  &c.  currency, 
o  federal  money.  Ans.  §761  42  cts.  7  m.+ 

11.  Change  219Z.  11s.  7$d.  New-England  and  Virginia 
currency,  to  federal  money.  Ans.  $731  94  cts.-{- 

*  A  dollar  is  8s.  in  this  currency — ,4=4-10  of  a  pound ;  therefore,  multi- 
ly  by  10.  and  divide  by  4,  brings  the  pounds  into  dollars,  &c. 


84  REDUcfio:;  OF  CUIUIESLILS. 

12.  Change  241Z.  New-England,  &c.  currency,  into  le* 
deral  money.  Ans.  $803, 33  ct*. 

13.  Bring  20/.  185.  5f<7.  New-England  currency,  into 
dollars.  Ans.  $69,  74  cts.  6|  m.+ 

14.  Reduce  468?.  New-York  currency  to  federal  money. 

Ans.  $1170* 

15.  Reduce  17s.  9j<f.  New- York,  &c.  currency,  to  dol- 
lars, &c.  Ans.  §2,  22  cts.  6,5  nt.+ 

16.  Borrowed  10  English  crowns,   at  6s.  Sd.  each,  how 
many  dollars,  at  6s.  each,  will  pay  the  debt? 

Ans.  $11,  11  cts.  1  m. 

NOTE. — There  are  several  short  practical  methods  of  re- 
ducing New-England  and  New- York  currencies  to  Federal 
Money,  for  which  see  the  Appendix. 

CASE  IF. 
To  reduce  the  currency  of  New- Jersey,  Pennsylvania. 

Delaware,  and  Maryland,  to  Federal  Money. 
RULE. — Multiply  the  given  sum  by  0,  and  divide  tho  pro'ducl 
and  the  quotient  will  be  dollars,  &c.* 

EXAMPLES. 

1.  Reduce  245/.  New-Jersey,  &c.  currency,  to  fed 
money. 

£245x8=1960,  and  19t>0-^3==$653j=$658,  33-£cfc» 

NOTE. — When  there  are  shillings,  pence,  &c.  in  the  eivqa 
sum,  reduce  them  to  the  decimal  of  a  pound,  then  multiply 
and  divide  as  above,  &c. 

2.  Reduce  36J.  11s.  8±d.  New-Jersey,  <fcc.  currency,  to 
federal  money.       £36,5854  decimal  value. 

8 

3)292,6832(97,56106  Ans.     ANSWER?, 

£,.  s.  d.  $     cts.  m. 

3.  Reduce  240  0  0  to  federal  money  640  00 

4.  Reduce  125  8  0 

5.  Reduce     99  7  61  265  00  5+ 

6.  Reduce  100  0  0^  266  66  6-f 

7.  Reduce     25  3  7  67  14  4 

8.  Reduce       0  17  9                                      2  36  6,6 

*  A  dollar  is  7s.  Gd.=90d.  in  this  currency=00-240=^3-8  of  a  pound  ;  1 
for*1,  multiplying  by  8.  and  dividing  by  3,  gives  the  dollars, 


T  3, 


REDUCTION  OF  CURRENCIES.  80 

CASE  III. 
To  reduce  the  currency  of  South-Carolina  and  Georgia, 

to  Federal  Money. 

RULE. — Multiply  the  given  sum  by  30,  and  divide  the  product  by 
',  ihe  quotient  will  be  the  dollars,  cents,  &LC.V 

EXAMPLES. 

1.  Reduce  100/.  South-Carolina  and  Georgia  currency, 
o  federal  money. 

100Z.  x  30^,3000  ;  3000     7^428,5714  Arts. 

2.  Reduce  54/.   16s.   9|d.   Georgia  currency,  to  federal 
uionev.  54,840t>  declined  expression. 

30 


7)1645,2180 

AKS.  285,0311 

£        s,      d. 
3.  Reduce    94     14     8  to  federal 
A    Rpflnpo      1Q      17      ft* 

ANSWERS. 

S     cts.  m. 
money,    405  99  8+ 
cx   ix  74- 

^    l?n/]iif»n  417      1-1      ft" 

170H   9^ 

G~p4>f1ii"r>    1,1ft        10       f> 

1\£\?>     IJOi. 

7     R<>/]npj>    Iftfl          0       O 

ftQ.X    71     J.  >— 

SRf.'Iiipn          f>        11        ft 

O    1^     /fJ- 

0.  •RP.IM^P     41       17      0 

I7O   .^1    4  8 

CASE  IV. 
To  reduce  the   currency  of  Canada  and   Nova-Scotia  to 

Federal  Moitey. 

RULE. — Multiply  the  given  sum  by  4,  the  product  will  be  dollars. 
NOTE. — Five  shillings   of  this  currency  are  equal  to   ri 
dollar  ;  consequently  4  dollars  make  one  pound. 

EXAMPLES. 

1.  Reduce  1257.  Canada  and  Nova-Scotia  currency,  to 
federal  money.  125 

4 

Ans.  $500 


•    4s.   Sd.    or  56f7.   to  the  dollar— -x5^—^  nf  a  pound  ; 
00-?-7, 


80  KKDUCTION  OF  COltf. 

2.  Reduce  55?.  10s.  6d.  Nova-Scotia  currency,  to  dollar?, 

55,525  decimal  value. 
4 

--        I    cts. 

Ans.  $222,  100=r222  10  ANSWERS'. 

3.  Reduce  241/.  18s.  9J.  to  federal  money,  £007  75 

4.  Reduce    58    13    6J  234  fO 

5.  Reduce  528    17    8  2115  53 
G.  Reduce      120  4  50 

7.  Reduce  224    19     0  -  —  899  80 

8.  Reduce      0    13  Hi  2  79 

REDUCTION  OF  COIN. 

RULES  for  reducing  the  Federal  Money  to  the  currencies 

of  the  several  United  States. 
To  reduce  Federal  Money  to  the  currency  of 

RuLE._Multiply  the  given  sum  by  ,3,  and  tho 
I.    <  j-?^™1,0'         ,>       product  will  be  pounds,  and  decimals  of  a 


)  RULE.  —  Multiply  the  given  sum  by  ,4,  and  thfj 

\      product  will  be  pounds-,  and  decimals  of  a 


LK.—  Multiply  the  given  sum  by  ,3,  and  di~ 
Pennsylvania^  I       vide  the  prot]uct  by  8,  and  the  quotient  will 
Delawarejondi       bc  poundSi  and  decimals  of  a  pound. 
Maryland.        j 

South  r<firn1inf,  )  RULE.—  Multiply  the  given  sum  by  ,7  and 

and  T        divide    by  3,    the    quotient  will    be  the 

Georgia         C        answer    in    pounds,  and    decimals   of  a 

3        pound. 

Examples  in  the  foregoing  Rules. 
.  Reduce  $152,  60  cts.  to  New-England  currency* 
,3 


£45,  780  ^ws.=£45  15s. 

20  But  the  value  of  any  decimal  of 
--  a  pound,  may  be  found  by  inspec- 
15,  600  tion.  See  Problem  IJ.  page  81. 


200 


REDUCTION  OF  COIN*.  67 

•2.  In  $196,  how  many  pounds,  N.  England  currency  t 
__  ._fl 

£58,8  Ans.=£5S  16 

3.  Reduce  $629  into  New-  York,  &c.  currency. 
,4 


_ 

£-251,6  ^4ns.~£251  12 

4.  Bring  SI  10,  51cts.  1m.  into  New-  Jersey,  &c.  currency. 
#110,511 

,3         Double  4  makes  8s.    Then  39  farthings 
8)331,533    are  9d.  3qrs.    See  Problem  II.  page  81. 
" 


"£11,441  ^4ns.=:£41  8s.  9J<7.  %  Inspection. 

.  Brin*  $65,  36cts.  into  South-Carolina,  &c.  currency. 


3),45, 752 

£15,250— £15  5s.  Ans.  ANSWERS. 

$     cts.  £      s.     d. 

6.  Reduce  425,07  to  N.  E.  &c,  currency.  127  10     5  + 

7.  Reduce    36,11  to  N.  Y.  &c.  currency.     14     8  10|4- 

8.  Reduce  315,44  to  N.  J.  &c.  currency.  118     5     9|+ 

9.  Reduce  690,45  to  S.  C.  &c.  currency.  161     2    1,2 

To  reduce  Federal  Money  to  Canada  and  Nova-  Scotia  currency. 
RULE. — Divide  the  dollars,  £c.  by  4,  the  quotient  will  be  pounds, 
ind  decimals  of  a  pound. 

EXAMPLES. 

1.  Reduce  $741  into  Canada  and  Nova-Scotia  currency. 
S    cts. 


~£185,25=£185  5s. 

2.  Bring  $311,  75  cts.  into  Nova-Scotia  currency. 

$  cts. 
4)311,750 
£77,9375—  £77  18s.  9d. 

3.  Bring  #2907,  56  cts.  into  Nova-Scotia  currency. 

Ans.  £726  17s. 

4.  Reduce  $2114,  50  cts.  into  Canada  currency. 

Ans.  £528  12$,  Orf, 


<ot>  RULES  FOR  REDUCING  CURRENCIES. 

RULES  far  reducing  the  Currencies  of  the  several  United  States,  aiao 

Canada,  JVova  Scotia,  and  Sterling,  to  the  par  of  all  the  others. 
tCjP*  See  the  given  currenc)  in  the  left  hand  column,  and  then  cast  your 
eye  to  the  right  hand,  till  you  come  under  the  required  currency,  and  you 
will  have  the  rule. 


MagMmcoBw 

Jfew-Ene- 
and,      yir- 
ffinia,  Km- 
ucky,    and 
renncsuce. 

NewJersey, 
Pennsylva- 
nia, Dcla- 
irnre,  and 
Maryland. 

JVw  York, 
and  Jforth- 
Carolina. 

South-  Ca- 
rol in  a,  and 
Georgia. 

Canada, 
and 
NovaScotia, 

Sterling. 

*\*t  to-  Eng- 
land, I'ir- 
pinia,  Ken- 
tucky, and 
yjfnnessce. 

Add  one 
fourth  to  the 
given  sum. 

Add  one 
third  to  the 
given  sum. 

Multiply  the 
i^iven  sum 
tjy  7,  and  di- 
vide the  pro- 
duct by  », 

Multiply  the 
given  sum 
by  5,  and  di- 
vide the  pro- 
duct by  6. 

Deduct  one 
ouilh  from 
he  givtn 
sum. 

New  Jersey, 

Pennsylva- 
nia, Dela- 
jrnre,  and 
Maryland. 

Deduct  one 
fifth       from 
he       given 
aum. 

Add  one 
fifteenth  to 
the  given 
sum. 

Multiply  the 
Riven  sum 
by  28,  and 

livido  the 
product  by 

Deduct  one 
third  from 
the  given 
sum. 

Vuli'ply  the 
§iv»  n      sum 
y  3,  iuid  di- 
vide the  pro- 
duct by  5. 

Mic-  York, 
and  Jfortft- 
Caiolina. 

Deduct  one 
fourth    from 
the      Ncw- 
York,  &c. 

Deduct  one 
sixteenth 

from  the  N. 
York. 

Multiply  the 
sjivcn  sum 
by  7,  and  di- 
vide the  pro- 
duct by  12. 

Multiply  the 
?iven  sum 
By  5,  and  di- 
vide the  pro- 
duct by  8. 

Multiply  the 
;iven  sum 
jy  9,  and  di- 
vide the  pro- 
duct by  16. 

South-  Ca 
rstliiia,    and 
Georgia. 

Multiply  thg 
§iven     sum 
y  9,  and  di- 
vide the  pro- 
duct by  7. 

Multiply  the 
given  sum 
by  45,  and 
divide  the 
product  by 
28. 

Multiply  the 
given  sum 
by  12,  and 
divide  the 
product  by  7. 

Multiply  the 
given  Btiir 
by  15,  and 
divide  tht 
product  by 
14. 

From  the 
given     sum 
deduct    one 
twenty- 
eighth. 

Canada, 
and 
WovaScotia 

Add     one 
fifth    lo  the 
Canada.&c 

Add    one 
half  to    the 
Canada 
sum.*' 

Multiply  the 
riven  sum 
by  8,  and  di- 
vide the  pro- 
duct by  5.« 

Deduct  one 
fifteenth 
from  the  gi- 
ven sum. 

Deduct  one 
tenth  from 
the  eiven 

yum. 

Sterling. 

To  the  En- 
glish     sum 
add         one 
third. 

Multiply  the 
English  mo- 
ney by  5,  and 
divide  the 
product  by3. 

Multiply  the 
English  sum 
by  10,  and 
divide  the 
1  product  by9. 

To  the  Eng 
ish    .money 
add         one 
twenty-se- 
venth. 

Add  one 
ninth  to  the 
given  sum. 

REDUCTION  OF  CulX.  % 

Implication  of  the  Rules  contained  in  the  foregoing  Table. 

EXAMPLES. 

1.  Reduce  46/.  10s.  6d.  of  the  currency  of  New-Hamp- 
hire,  into  that  of  New-Jersey,  Pennsylvania,  &c. 

£.    s.    d. 

See  the  rule  4)46  10  6 

in  the  table.  +11   12  7J 


Ans.  £58    3  H 

2.  Reduce  25/.  13s.  9d.  Connecticut  "currency,  to  New- 
fork  currency.  £.    s.    d. 

3)25  13  9 
By  the  table,  +£,  &c.  -f-8  11  3 

Ans.  £34     5  0 

3.  Reduce  1257.  10g.  4d.   New-York,  &c.  currency,  to 
South-Carolina  currency.         £.    s.    d. 

Rule  by  the  table,  125  10  4 

x7,-rby  12,  &c.  7 

12)878  12  4 


Ans.  £73     4  4£ 

4.  Reduce  467.  lls.  8d.  New-York  and  N.  Carolina  cur- 
•ency,  to  sterling  or  English  money.         £    s.    d. 

4G  11  8 
9 


See  the  table.  \  IG^X  4)419     5  0 
X  given  sum  by  >  4)104  16  3 

9,-:-byl6,&c.     )  

Ans.  £26     4  OJ 

To  reduce  any  of  the  different  currencies  of  the  several 
States  into  eacli  other,  at  par ;  you  may  consult  the  prece- 
ding1 table,  which  will  give  you  the  rules. 

MORE  EXAMPLES  FOR  EXERCISE. 

5.  Reduce  84/.  10s.  8d.  New-Hampshire,  &c.  currency, 
into  New- Jersey  currency.  Ans.  £105  13s.  4d. 

6.  Reduce  120?.  8s.  3d.  Connecticut  currency,  into  New- 
York  currency.  Ans.  £160  lls.  M. 

H  2 


430  RULE  OF  THREE  DIRECT. 

7.  Reduce  120Z.  10s.  Massachusetts  currency,  into  South- 
Carolina  and  Georgia  currency.  Ans.  £93  14s.  5]</. 

8.  Reduce  4107.   18s.  lid.  Rhode-Island  currency,  into 
Canada  and  Nova-Scotia  currency.       Ans.  £342  95.  Id. 

9.  Reduce  524/.  8s.  4d.  Virginia,  &c.  currency,  into  ster- 
ling money.  Ans.  £393  6s.  3d. 

10.  Reduce  214/.  9s.  2d.  New-Jersey,  &c.  currency,  into 
N.  Hamp.  Massachusetts,  &c.  Qiirrency.  'Ans.  1711. 1  Is.  4d. 

11.  Reduce  100/.  New-Jersey,  &c.  currency,  into  New- 
York  and  North-Carolina  currency.     Ans.  106/.  13s.  4</. 

12.  Reduce  100/.  Delaware  and  Maryland  currency  into 
sterling  money.  Ans.  CO/. 

13.  Reduce  116Z.  10s.  New-York  currency,  into  Connec- 
ticut currency.  ^l??s.  871.  7s.  Gd. 

14.  Reduce  112£.  7s.  3d.  S.  Carolina  and  Georgia  curren- 
cy, into  Connecticut,  &c.  currency.     Ans.  1447.  9s.  3f  d. 

15.  Reduce  100/.  Canada  and  Nova-Scotia  currency,  in 
Connecticut  currency.  Ans.  1207. 

16.  Reduce  116/.  14s.  9d.  sterling  money,  into  Connec- 
ticut currency.  Ans.  1551.  13s. 

17.  Reduce  104/.  10s.  Canada  and  Nova-Scotia  curren- 
cy, into  New- York  currency.  Ans.  1677.  4s. 

18.  Reduce  1007.  Nova-Scotia  currency,  into  New-Jer- 
sey, &c.  currency.  Ans.  150/. 


RULE  OF  THREE   DIRECT. 

THE  Rule  of  Three  Direct  teaches,  by  having  three 
numbers  given  to  find  a  fourth,  which  shall  have  the  same 
proportion  to  the  third,  as  the  second  has  to  the  first. 

1.  Observe  that  two  of  the  given  numbers  in  your  ques- 
tion are  always  of  the  same  name  or  kind  ;   one  of  which 
must  be  the  first  number  in  stating,  and  the  other  the  third 
number  ;  consequently  the  first  and  third  numbers  must  al- 
ways be  of  the  same  name,  or  kind  ;-and  the  other  number, 
which  is  of  the  same  kind  with  the  answer,  or  thing  sought, 
will  always  possess  the  second  or  middle  place. 

2.  The  third  term  is  a  demand  ;  and  may  be  known  by 
these  or  the  like  words  before  it,  viz.  What  will  ?  What  cost? 
How  many  ?  How  far?  How  long?  or.  How  much?  &c. 


RULE  OF  THREE  DIRECT.  9i 

RULE. — 1.  State  the  question  ;  that  is,  place  the  numbers  so  that 
the  first  and  third  terms  may  be  of  the  same  kind  ;  and  the  second 
term  of  the  same  kind  with  the  answer,  or  thing1  sought. 

2.  Bring  the  iirst  and  third  terms  to  the  same  denomination,  and 
reduce  the  second  term  to  the  lowest  name  mentioned  in  it. 

3.  Multiply  the  second  and  third  terms  together,  and  divide  their 
product  by  the  first  term  ;  and  the  quotient  will  be  the  answer  to  the 
question,  in  the  same  denomination  you  left  the  second  term  in,  which 
may  be  brought  into  any  other  denomination  required. 

The  method  of  proof  is  by  inverting  the  question. 
[NOTE. — The  following  methods  of  operation,  when  they  can  be  used, 
perform  the  work  in  a  much  shorter  manner  than  the  general  rule. 

1 .  Divide  the  second  term  by  the  first ;  multiply  the  quotient  into  the  third, 
and  the  product  will  be  the  answer.     Or, 

2.  Divide  the  third  term  by  the  first ;  multiply  the  quotient  into  the  second, 
and  the  product  will  be  the  answer.     Or, 

3.  Divide  the  first  term  by  the  second,  and  the  third  by  that  quotient,  and 
the  last  quotient  will  be  the  answer.     Or, 

4.  Divide  the  first  term  by  the  third,  and  the  second  by  that  quotient,  and 
the  last  quotient  will  be  the  answer.] 

EXAMPLES. 

1.  If  6  yards  of  cloth  cost  9  dollars,  what  will  20  yards 
cost  at  the  same  rate  ?  Yds.  $          Yds. 

Here  20  yards,  which  moves  the  6:9    :  :    20 

question,  is  the  third  term;  6yds.  9 
the  same  kind,  is  the  first,  and  9 

dollars  the  second.  6)180 

Ans.  $30 

2.  If  20  yards  cost  30  dols.         3.  If  9  dollars  will  buy  6 
what  cost  6  yards  1  yards,  how  many  yards  will 

Yds.     $  Yds.  30  dols.  buy]  $  yds.      $ 

20  :  30  :  :  6  9  :  6  :  :  30 

6  6 

2,0)18,0  9)186 


MS-  $9  Ans.  20^7. 

4.  If  3  cvvt.  of  sugar  cost  8/.  8s.  what  will  11  cwt.  I  qr. 
24  Ib.  cost  1 

3  cwt.   £/.  Ss.     C.  qr.  Ib.  Ib.         s. 

112         20  11    1    24     As  336  :   168::  1284/6. 

IS*.  _^! 

45           [Carried  up.]  10272 


HULE  0?  THIIEK  DIRECT, 

45  10:27-2 

28  7704 

1284 

364  (2,0) 

92  336)215712(64,2 

2016 

1384  32/.2s. 

141]      A->s. 
1344 

"672 
672 

o.  If  one  pair  of  stockings  cost  4s.  6d.  what  will  19  do- 
zen pair  cost  ?  Ans.  £51  6s. 

6.  If  19  dozen  pair  of  shoes  cost  51Z.  6s.  what  will  one 
pair  cost  ?  Ans.  4s.  6 d. 

7.  At   I OU1.  per  pound,  what  is  the  value  of  a  flr'dn  of 
butter,  weight  50  pounds?  Ans.  £2  9s. 

8.  How  much  sugar  can  you  buy  for  23/.  2s.  at  9d.  per 
pound  I  Ans.  5  C.  2  qrs. 

9.  Bought  8  chests  ^f  sugar,  each  9  cwt.  2  qrs.  what  do 
they  come  to  at  2/.  5s.  per  cwt.  1  Ans.  £171. 

10.  If  a  man's  wages  be  75/.  10s.  a  year,  what  is  that  a 
calendar  month?  Ans.  £6  5s.   Wd. 

11.  If  4;  tuns  of  hay  will  keep  3  cattle  over  the  winter; 
how  many  tuns  will  it  take  to  keep  25  cattle  the  same  time? 

Ans.  37 J  tuns. 

12.  If  a  man's  yearly  income  be  2087.  Is.  what  is  that  a 
day?  Ans.  Us.  4</.  o^-fy  qrs. 

13.  If  a  \in\\\  gpcr.cl  3s.  4d.  per  day,  how  r.»nch  is  that  a 
year?  Ans.  £60  Ws.Sd. 

14.  Boarding  at   12s.  (xl.  per  week,  how  long  wrill  32J. 
10s.  last  me  ?  A?is.  1  year. 

15.  A  ov/c;s  B  3475Z.  bat  B  compounds  with  him  for  13s. 
4d.  on  the  pound  ;  pray  what  must  he  receive  for  his  debt? 

Ans.  £2316  135.  U. 

16.  A  goldsmith  sold  a  tankard  fi/r  SI.  12s.  at  5s.  4d.  per 
oz.whnt  was  the  weight  of  the  tankard?  Ans.  2  Ib.  8  oz.  5pwt. 

17.  If  2  cwt.  3  qrs.  21  Ib.  of  susfar  cost  6/.  Is.  8d.  what 
cost  35}  cwt ,  ?  Ans.  £73, 


RULE  OF  THREE  DIRECT.  93 

18.  Bough':  10  pieces  of  cloth,  each  piece  containing  9J 
yards,  at  1  Is.  4M.  per  yard  ;  what  did  the  whole  come  to  1 

Ans.  £55  95.  Q*d. 

FEDERAL  MONEY. 

jNViE  1.     You  must  state  the  question,  as  taught  in  the 
foregoing,  and  after  reducing  the  first  and  third  terms 
Vo  tie  same  name,  &c.  you  may  multiply  and  divide  accord- 
ing to  the  rules  in  decimals  ;  or  by  the  rules  for  multiplying 
and  dividing  Federal  Money. 

EXAMPLES. 

19.  If  7  yds.  of  cloth  cost  15  dollars  47  cents,  what  will 
12yds.  cost?  Yds.     $  cts.       yds. 

7  :   15,47  :  :   12 
12 


7)185,64 

Ans.  26,52=:$26,  52  cts. 

But  any  sum  in  dollars  and  cents  may  be  written  down 
as  a  whole  number,  and  expressed  in  its  lowest  denomina- 
tion, as  in  the  following  example:  (See  Reduction  of  Fede- 
ral Money,  page  62.) 

20.  What  will  1  qr.  91b.  sugar  come  to,  at  6  dollars  45 
Cts.  per  cwt.  ?  qr.  Ib.  Ib.  cts.  Ib. 

1     9    As  1 12  :  645  :  :  37 
28  37 


37  Ib.        4515 
1935 

cts. 

1 12)23865(213  +  Ans=$2,\3. 
224 

146 
112 

345 
336 


94  RULE  Of  THREE  DIRK- 

NOTE  2.  When  ths  first  and  third  numbers  arc  federal 
money,  you  may  annex  ciphers,  (if  necessary,)  until  you 
make  their  decimal  places  or  figures  at  the  right  hand  of 
the  separatrix,  equal :  which  will  reduce  them  to  a  like  de- 
nomination. Then  you  may  multiply  and  divide,  as  in  whole 
numbers,  and  the  quotient  will  express  the  answer  in  the 
least  denomination  mentioned  in  the  second,  or  middle  terra. 
EXAMPLES. 

21.  If  3  dols.  will  buy  7  yds.  of  cloth,  how  many  yds.  can  I 
buy  for  120  dols.  75  cts.?     cts.       yds.         cts. 

As  300    :    7  :    :  12075 
7 

300)8452of281£  Ans. 

22.  If  12  Ib.  of  tea  cost  6  dols.  600 
78  eta.  and  9  mills,  what  will  5  Ib. 

cost  at  the  same  rate  ?  2452 

Ib.     mills.       Ib.  2400 

As  12  :  67S9  :  :  5 

5  525 

. 300 

12)33945 

fycts.m. 

Ans.  2828+w«7/s.— 2,82,8. 

900(3  qrs. 
900 
$      cts. 

23.  If  a  man  lay  out  121,  23  in  merchandise,  and  thereby 
gains  $3951  cts.  how  much  will  he  gain  by  laying  out  ,A1 
at  the  same  rate  1         Cents.       Cents.         Cents. 

As  12123    :    3951    :    :    1200 
1200 

rt'i,        ft    CtS. 

12123)4741200(391=3,91  Ans, 
36369 

110430 

109107 

. —  [Carried  up.] 


£  OF  TH'Rfc'ii  IMUECT. 

13230 
12123 


1107 

24.  If  the  wages  of  15  nxeks  come  to  §6-1  19  cts.  what  Is 
i year's  wages  at  that  ru  Ans.  $222,  52 cts.  5m. 

"25.   A  men  bought   sheep  at  $1    11  cts.  per  head,  to  the 
imount  of  51  dols.  6  cts. ;  how  many  sheep  did  he  buy  ? 

-4ns.  46. 

26.  Bought  4  pieces  of  cloth,  each  piece  containing  31 
-ards,at  16s.  6d.  per  yard,  (New-England  currency;)  what 
Iocs  the  whole  amount  to  in  federal  money?    Ans.  §341. 

27.  When   a  tun  of  wine  cost  140  dollars,  what  cost  a 
part  ?  Ans.  13  cts.  SfTt  m. 

28.  A  merchant  agreed  with  his  debtor,  that  if  he  wouKl 
)ay  him  dowa  65  cts.  on  a  dollar,  he  would  give  him  up  a 
lote  of  hand  of  249  dols.  88  cts.    1  demand  what  the  debtor 
nu.*t  p-.iy  for  his  note?  Ans.  $162  42 cts.  2m. 

29.  If  12  horses  eat  up  30  bush,  of  oats  in  a  week, how  many 

i  serve  45  horses  the  same  tim<  ?  Ans.  1 12  J  bush. 
I  30.  Bought  a  piece  of  cloth  for  $48  27  cts.  at  $1  19  cts.  per 
"d. ;  how  many  yds.  did  it  contain?  AnsAQyds.  2  grs.-pfo. 

31.  Bought  3  lihds.  of  sugar,  each  weighing  8  cwt.  1  qr. 
2  Ib.  at  £7  26  cts.  per  cwt.  what  come  they  to  ? 

Ans'.  $182  let.  Sm. 

32.  What  is  the  price  of  4  pieces  of  cloth,  the  first  piece 
joutaining  21,  the  second  23,  the  third  24,  and  the  fourth 
!7  yards,  :\t  I  dollar  43  cents  per  yard? 

Ans.  $13585  cts.  21  f  23-4-24-f  27=95jwfc. 
;3.   Bought  3  hhds.  of  brandy,  containing  61,  62,  62£- 
.  -it  1  dollar  38  cts.  per  gallon,  I  demand  how  mucb 
hey  an-  >u:>t  to  ?  Ans.  $255  99  cts. 

34.  Suppose  a  gentleman's  income  is  $1835  a  year,  and 
:^  19  cts.  a  day,  one  day  with  another,  how  muck 
r.ll  !,v.  !i;ivc  saved  at  the  year's  end?         ,-i?;9.$562, 15  cts. 

35   Tr    >iy  horse   stond  me  in  20  cts.   per  day  keeping, 
rhat  will  be  the  charge  of  11  horses  for  the  vvW.  at  that 

Ans. 


96  RULE  OF  THiXEE 

36.  A  merchant  bought  14  pipes  of  wine,  and  is  allowe 
6  months  credit,  but  for  ready  money  gets  U  8  cts.  a  gallo 
cheaper  ;  how  much  did  he  save  by  paying  ready  money  ? 

Ans.  $141,  }%cts. 
Examples  promiscuously  placed. 

37.  Sold  a  ship  for  537?.  and  I  owned  f  of  her ;   wha 
was  my  purt  of  the  money  1  Ans.  £201  7s.  6d. 

38.  If  ,  y  of  a  ship  cost  781  dollars  25  cents,  what  is  the 
whole  v  ->rth?  As  5  :  781,25 :  :  16  :  $2500  Ans. 

39.  V    I   buy  54  yards  of  cloth  for  3U  10s.  what  did  it 
cost  p< :  Ell  English  1  Ans.  14s.  7d. 

40.  knight  of  Mr.  Grocer,  11  cwt.  3qrs.  of  sugar*,  at  8 
dollai     i2  cents  per  cwt.  and   gave  him  James  Paywell's 
note  for  19Z.  7s.  (New-England  ciirrtvicy)  the  rest  I  pay  in 
cash  ,  tell  me  how  many  dols.  will  make  up  the  balance  1 

Ans.  §30,  91  cts.   > 

41.  If  a  staff  5  feet  long  cast  a  shade  or.  level  ground  8 
feet,  what  is  the  height  of  that  steeple  whose  shade  at  th<j*j 
same  time  measures  181  feet?  Ans.  113^  /?. 

42.  If  a  gentleman  have  an  income  of  300  English  gui- 
neas a  year,  how  much  may  he  spend,  one  day  with  ano- 
ther, to  lay  up  §500  at  the  year's  end  ?  /ins.  >2,  46 cts.  5m. 

43.  Bouo-ht  51)  pieces  of  Lerseys,  each  34  Eils  Flemish,  at ' 
8s.  4d.  per  Ell  English;  what  did  the  whol^  cost?  Ans.  £425. 

44.  Bought  200  yards  of  camhriok  for  90/.  but  being  da* 
mnged,  I  am  willing  to  lose  7L  10?.  by  the  sale  of  it ;  what 
must  I  demand  per  EH  English?  Ans.  10s.  3fd.    • 

45.  How  many  pieces  of  Holland,  each  20  Ells  Flemish, 
may  1  have  for23/.8s.  at  6s^6d.  per  Ell  English?  .4ns.  6; 

46.  A  merchant  bought  a  bale  of  cloth  containing  240  yds. ' 
at  the  rate  of  £7^  for  5  ycb.  and  sold  it  again  at  the  rate  oi 
$1H  for  7  yards ;  did  he  gain  or  lose  by  the  bargain,  and  how 
much  ?  Ans.  He  gained  $25,  71  cts.  4  »:. 

47.  Bought  a  pipe  of  wine  for  84  dollars,  and  found  it  hat 
leaked  out  12  gals.  :  I  sold  the  remainder  at  12^  cts.  a  pint 
what  did  I  gain  or  lose  ?  Ans.  I  gained  $30. 

48.  A  gentleman  bought   18  pipes   of  wine   at  12s.  Gd 
(New-Jersey  currency)  per  gallon ;  how  many  dollars  wil 
pay  the  purchase  ?  An*-  $3760. 


49.  Bought  a  quantity  of  plate,  weighing  15  Ib.  11 02?.  13 
>\vt.  17  gr.  how  many  dols.  will  pay  for  it,  at  the  rate  of  12s* 
Td.  New-York  currency,  per  oz.?  Ans.  §301,  50,  cts.Sfym. 

50.  A  factor  bought  n  certain  quantity  of  broadcloth  and 
trugget,  which  together  cost  817.  the  quantity  of  broadcloth 
ivas  50  yds.,  at  18s.  per  yd.,  and  for  every  5  yds.  of  broad- 
cloth he  had  9  yards  of  drugget ;  I  demand  how  many  yds. 
}f  drugget  he  had,  and  what  it  cost  him  per  yard  ? 

Ans.  90  yds.  at  8s.  per  yd. 

51.  If  I  give  1  eagle,  2  dols.  8  dimes,2ets.  and  5m.for675 
lops,  how  many  tops  will  19  mills  buy  ?  Ans.  1  top. 

52  Whereas  an  eagle  and  a  cent  just  threescore  yards 

did  buy, 

How  many  yards  of  that  same  cloth  for  15  dimes  had  1 1 

Ans'.  8  yds.  3  qrs.  3  na.-\- 

53.  If  the  legislature  of  a  state  grant  a  tax  of  8  mills  on 
tire  dollar,  how  niuoh  must  that  man  pay  who  is  319  dols, 
75  ceuts  on  the  list  1  Ans.  $2,  55  cts.  S  m. 

54.  If  100  dols.  gain  6  dols.  interest  in  a  year,  how  much 
will  49  dols.  gain  in  the  same  time  ?         Ans.  §2,  94  cts. 

55.  If  60  gallons  of  water,  in  one  hour,  fall  into  a  cistern 
containing  300  gallons,  and  by  a  pipe  in  the  cistern  35  gal^ 
Ions  run  out  in  an  hour;  in  what  time  will  it  be  filled  ? 

Ans.  in  12  hours. 

56.  A  and  B  depart  from  the  same  place  and  travel  the 
same  road ;  but  A  goes  5  clays  before  B,  at  the  rate  of  15 
miles  a  day ;   B  follows  at  the  rate  of  20  mile  a  day  ;   what 
distance  must  he  travel  to  overtake  A?      Ans.  300  miles. 


RULE  OF  THREE  INVERSE. 

THE  Rule  of  Three  Inverse,  teaches  by  having  three 
numbers  given  to  find  a  fourth,  which  shall  have  the  same 
proportion  to  the  second,  as  the  first  has  to  the  third. 

If  more  requires  more,  or  less  requires  less,  the  question 
belongs  to  the  Rule  of  Three  Direct. 

But  if  more  requires  less,  or  less  requires  more,  the  ques- 
tion belongs  to  the  Rule  of  Three  Inverse  ;  which  may  al- 
ways be  known  from  the  nature  and  tenor  of  the  question « 
For  example  r 


UF   THREE 


If  2  men  can  mow  a  field  in  4  dnys>  how  many  day  3  vvifl 
it  require  4  men  to  mow  it? 

men  days  men 

1.  If  2    require    4    how  much  time  will    4    require? 
Answer,  2  days.      Here  more  requires  less,  viz.  the  more 

-men  the  less  time  is  required. 

men  days  men 

2.  If  4   require    2    how  much  time  will    2    require  ? 
Answer,  4  days.     Here  less  requires  more,  vizr.  the  less  tho 
number  of  men  arc,  the  more  days  are  required  —  therefore 
the  question  belongs  to  Inverse  Proportion, 

RULE.  —  1.  State  anil  reduce  the  terms  as  in  the  Rule  of  Three  Di- 
rect. 

2.  Multiply  the  first  and  second  terms  together,  and  divide  the  pro- 
tiuct  by  the  third  ;  the  quotient  will  he  the  answer  in  the  same  deno- 
mination as  the  middle  term  was  reduced  into. 
EXAMPLES. 

1.  If  12  men  con  build  a  wall  in  20  days,  how  many  men 
can  do  the  same  in  8  days?  Ans.  30  men. 

2.  If  a  man  perform  a  journey  in  5  dnys,  when  the  day 
is  12  hours  long',  in  how  many  days  will  he  perform  it  wlnm 
the  day  is  but  10  hours  long?  Ana.  6  days. 

3.  What  length   of  board   7  J   inches  wide,  will  make  a 
square  foot?.  Ans.  19.1  inches. 

4.  If  five  dollars  will  pay  for  the  carriage  of  2  cwt.  150 
miles,  how  far  may  15  cwt.  be  carried  for  the  same  money? 

Ans.  20  mifas. 

5.  If  when  wheat  is  7s.   6d.  the  bushel,  the  penny  Ion  f 
will  Weigh  9  oz.  what  ought  it  to  weigh  when  wheat  is  O's. 
per  bushel  ?  Ans.  11  oz.  5pwt. 

6.  If  30  bushels  of  grain,  at  50  cts.  per.  bushel,  will   pay 
a  debt,  how  many  bushels  at  75  cents  per  bushel,  will  pay 
the  same  ?  Ans.  20  bushels. 

7.  If  100Z.  in  12  months  gain  6L  interest,  what  principal 
will  gain  the  same  in  8  months  ?  Ans.  £150. 

8.  If  11  men  can  build  a  house  in  5  months,  by  working 
12  hours  per  day—  in  what  time  will  the  same  number  of 
men  do  it,  when  they  worfc  only  8  hours  per  day  ? 

Aiis.  7£  months. 

9.  What  number  of  men  must  be  employed  to  finish  ia 
,  -what  15  aaen^ould  bo  20  days  about? 


10.  buppose  050  men  art;  in  a  garrison,  and  their  provi- 
sions calculated  to  last  but  2  months,  how  many  men  must 
leave  the  garrison  that  the  same  provisions  may  be  suffi- 
cient lor  those  who  remain  5  months  'I         Ans.  390  men. 

11.  A  regiment  of  soldiers  consisting  of  850  men  are  to 
be  clothed,  each  suit  to  contain  3^  yards  of  cloth,  which  is 

?  1 J  yds.  wide,  and  lined  with  shalloon  £  yd.  wide ;  how  ma- 
ny yards  of  shalloon  will  complete  the  lining? 

Ans.  6911  yds.'S  qrs.  2J  na. 

PRACTICE. 

PRACTICE  is  a  contraction  of  the  Rule  of  Three  Direct, 
Jpvhen  the  first  term  happens  to  be  a  unit  or  one,  and  is  h 
concise  method  of  resolving  most  questions  that  occur  in 
trade  or  business  where  money  is  reckoned  in  pounds,  shil- 
lings and  pence ;  but  reckoning  in  federal  money  will  ren- 
der this  rule  almost  useless  :  for  which  reason  1  shall  not 
tmlarge  so  much  on  the  subject  as  many  oilier  writers  have 
dune. 

Tables  of  Aliquot,  or  Even  Pa/'tis. 


Parts  of  a  shilling 
d.  s. 

0      is      -J- 

4  =      | 

5  ? 


Parts  of  2  shillings. 
i  i    ° 

Is.     is    £ 

8d.    =    1 
6d.  -> 

4*1-  * 


Parts  of  a  pound, 


s.  d. 

10  0 

G  8 

5  0 

4  0 

3  4 

2  (> 

1  S 


'  £ 


Parts  of  a  cwt. 
Ib.          cwt. 
r>G      is    -.>- 

2S       r^=      -1 

10 
14 


3d. 


The  aliquot  part  of  any  number  is 
such  a  part  of  it,  as  being  taken  a  cer- 
tain number  of  times,  exactly  mnkcb 
that  number. 


Ta 


CASE  I. 

When  the  price  of  one  yard,  pound,  &£.  is  an  oven  part 
of  one  shilling — Find  the  value  of  the  given  quantity  at 
Is.  a  yard,  pound,  &c,  and  divicl*  it  by  that  even  part,  and 
?,ho  rpmfjPT't  will  ].«»•  t-'.c  n*r»?Vv'.-r  in'  •  sVf 


100 

Or  find  the  value  of  the  given  quantity  at  2s.  per  yd.  &e. 
and  divide  said  value  by  the  even  part  which  the  given 
price  is  of  2s.  and  the  quotient  will  be  the  answe'r  in  shiT- 
lmtvs,&c.  which  reduce  to  pounds. 

N.  B.  To  find  the  value  of  any  quantity  at  Ss.  you  need 
only  double  the  unit  figure  for  shillings  ;  tire  other  figures 
will  be  pounds. 

EXAMPLES. 

1.  What  will  46]  i  yds.  of  tape  come  to  at  l£d.  per  yd.  ? 

s.     d. 
l|d.  |  |  |  461  6  value  of  461^  yds.  at  Is.  per  yd. 

5,7    8| 

£2  17s.  8J«7.  value  at  1-Jd. 

2.  What  cost  256  Ib.  of  cheese  at  8d.  pe*r  pound? 
ScL  j  -J-  |  £25  12s.  value  of  256  Ib.  at  2s.  perlh. 

£8  10s.  3d.  value  at  8d.  per  pound. 
Yards,  per  i/ard>  £.     s.  d. 

486£  at  Id.  Answers.  2    0  6j 

8G2     at  2d.  7    3  8 

Oil     at  3d.  II     7  9  . 

749     at  4d.  12     9  8 

113     at  6d.  2  16  6 

899    at'  8d.  29  19  4 

CASE  If. 

When  the  price  is  an  even  part  of  a  pound — Find  the 
value  of  the  given  quantity  at  one  pound  per  yard,  «fcc.  and 
divide  it  by  that  even  part,  and  the  quotient  will  be  tlte  an- 
swer in  pounds. 

EXAMPLES. 

What  will  129i-  yards  cost  at  2s.  6d.  per  yard  ? 
5v  d.  £.     9.  £. 

2  6  ]  -J-  |    129  10  value   at  I  per  yard. 

Ans.  £16  2s.  &d.  value  at  2s.  6d.  per  yank 
Y(fs.          5.  €?.  £.     &.  d. 

T23    at  10  0  per  van}.  An  SWA.  61   10  () 

6874  at     5  0       ,  I  171  17  ft 


I'KA-    .      ^. 

\tif.          &  d..  £.      s.  a, 

21 1J  at     4  0  per  yard.  42     5-0 

543    at     6  8       —  l$l     0  0 

127    at    3  4      -  -  21     3  4 

461    at     1  8      —  38     8  4 

NOTE. — When  the  price  is  pounds  ouly,  the  given  quan- 
ity  multiplied  thereby,  will  be  the  answer. 
EXAMPLE. — 11  tuns  of  hay  at  4Z.  per  tun.      Thus,  11 

4 

Ans.  £44 
CASE  III. 

When  the  given  price  is  any  number  of  shillings  un- 
der 20. 

1.  When  the  shillings  are  an  even  number,  multiply  the 
quantity  by  half  the  number  of  shillings,  and  double  the 
first  figure  of  the  product  for  shillings;  and  the  jest  of  the 
product  will  be  pounds. 

2.  If  the  shillings  be   odd,  multiply  the  quantity  by  the 
whole  number  of  shillings,  and  the  product  will  be  the  an- 
swer in  shillings,  which  reduce  to  pounds. 

EXAMPLES. 

1st.— 124  yds.  at  Sg.  3d.— 132  yds.  at  7s.  per  vd. 

4*  7 

£49  12s.  Ans.  2,0)92,4 

£46,4  Ans. 

Yds.  £.    s.     Yds.  £.     s. 

562  at     4s.       Ans.112    8  j  372  at  11*.       An*.  204  12 
378  at    2s.  37  16 1  264  at    Ps.  118  16 

913  at  14s.  630    2  |  250  at  16s.  200  00 

CASE  IV. 

When  the  given  price  is  pence,  or  pence  and  farthings, 
and  not  an  even  part  of  a  shilling — Find  the  value  of  the 
given  quantity  at  Is.  per  yd.  &c.  which  divide  by  the  great- 
est even  part  of  a  shilling  contained  in  the  given  price,  and 
take  parts  of  the  quotient  for  the  remainder  of  the  price, 
and  the  sum  of  these  several  quotients  will  be  the  answer 
•s.  Are,  which  reduce  to  pound?. 

,   i> 


•    .  ;  - 


EXAMPLES. 

Wha!  will  24511).  of  raisins  come  fo«  at  9|d.  per  Ik 

s.      d. 
6d.     %  '  245     0  value  of  245  Ib.  at  Is.  per  ; 

3d.     •£     122     6  value  of  do.  at  6d.  per  Ib. 

fd.     J       61     3  value  of  do.  at  3d.  per  Ib. 

15     3  £  value  of  do.  at  Jd.  per  Ib. 


Ib. 


2,0)19,9     Of 

Ans.  £9  19  Oj  value  of  the  whole  at  9|d.  per  Ib. 
d.  £.  a.      d.          Ib.  d.  £.     s.  d. 


372  at  1£     Ans.  2  14     3 
325  at  2}  30  11} 

§27  at  4}  15  10     H 


18    0  0 
20  17  0{ 

32  18  0 


576  at     7£ 
541  at     9} 
672  at  11 J 
CASE  V. 

When  the  price  is  shillings,  pence  and  farthings,  and  pot 
the  aliquot  part  of  a  pound— Multiply  the  given  quantity 
f>y  the  'shillings,  and  take  parts  for  the  pence  and  farthings, 
as  in  the  foregoing  cases,  and  add  them  together ;  the  sum 
will  be  the  answer  in  shillings. 

EXAMPLES. 
1.  What  will  246  yds.  of  velvet  come  to,  at  7s.  3d.  per  yd.'? 

s.     d. 
3d.  |  -J-  |  246  0  value  of  346  yiards  at  Is.  per  yd. 

1722  0  value  of  do.  at  7s.  per  yard. 
61  6  value  of  do.  at  3d.  per  yard. 


2,0)178,  3  6 
An».  £89  3  6  Value  of  <fo.  at  7s.  3d.  per  yard, 


d. 


2.  What  jcost  139  yds.    at     9  10  per  yd.  ? 

3.  What  cost  146  yds.    at  14    6  per  yd,? 

4.  What  cost  120  cwt. 

5.  What  Cflst  127  yds. 
a  What  cost     49J  Ibs. 


at  11  3  percwt.  ? 
at  9  Sjperyd.? 
at  SlUperlb.? 


ANSWERS. 
£.     s.    d. 

68    6  10 
107  13     6 
67  10 
12  1H 


61 


9  15  111 


103 


CASE  VI. 

When  the  piicc  and  quantity  given  are  of  several  deno- 
ainations — Multiply  the  price  by  the  integers  in  the  given 
juantity,  and  take  parts  for  the  rest  from  the  price  of  an  in- 
eger ;  which,  added  together,  will  be  the  answer, 
applicable  to  federal  money. 

EXAMPLES. 


This  is 


1.  What  cost  5  cwt.  3  qrs. 

2.  What  cost  9  cwt.  1  qr. 

4  Ib.  of  raisins,  at  2/.  11s. 

8  Ib.  of  sugar,  at  8  dollars, 

Hi.  per  cwt.  ? 

65  cts.  per  cwt.  1 

£. 

s.      d. 

$  cts. 

|    2  qrs. 

i 

2 

11     8 

Iqr. 

\ 

8,65 

[  i 

5 

9 

11 

12 

18     4 

77,85 

Iqr. 

i 

1 

5  10 

71b. 

t 

2,1625 

14  Ib. 

i 

12  11 

lib. 

1 

T 

,5406 

6    5£ 

,772 

Ans.  £15 

3   6J 

Ans.  $80,6303 

C.  qrs.  Ib. 

ANSWERS. 

7     3     16  at 

$9,  58  cts.  per  cwt.         $75,  61  cts.  3  m. 

51       0  at 

ZL  17s.  per  cwt.                   £14  19*.  3d. 

14     3       7  at 

07.  13s.  8d.  per  cwt.             £10  2s.  5\d. 

12     0      Tat 

$6,  34  cts.  per  cwt.        $76,  47  cts.  6  m. 

0     0    24  at 

$1  1,  91  cts.  per  cwt.     #2,  55  cts.  2TV  m. 

TARE  AND  TRET. 

TARE  and  Tret  are  practical  rules  for  deducting  cer- 
ain  allowances  which  are  made  by  merchants,  in  buying 
nd  selling  goods,  &c.  by  weight ;  in  which  are  noticed  the 
t)llowii>g  particulars  : 

1.  Gross  Weight,  which  is  the  whole  weight  of  any  sort 
f  goods,  together  with  the  box,  cask,  or  bag,  &c.  which 
ontains  them. 

.  Tare,  which  is  an  allowance  made  to  the  buyer,  for 
be  weight  of  the  box,  cask,  or  bag,  &c.  which  contains  the 
;oods  bought,  and  is  either  at  so  much  per  box,  &c.  or  at 

much  per  cw.t.  or  at.  sr>  much  in  the  whole  gross  weight. 


3.  Tret,  which  is  an  allowance  of  4  lh.  on  every  lOlll, 
for  waste,  dust,  &c. 

4.  Cloff,  which  is  an  allowance  made  of  2  Ib.  upon  ever,y 
3  cwt. 

5.  Buttle,  is  what  reflnains  after  one  or  two  allowances 
have  been  deducted. 

CASE  I. 

When  the  question  is  an  Invoice — Add  the  gross  weights 
into  one  sum  and  the  tares  into  another  ;  then  subtract  the 
total  tare  from  the  whole  gross,  and  the  remainder  will  he 
the  neat  weight. 

EXAMPLES. 

1.  What  is  the  neat  weight  of  4  hogsheads  of  Tobacco, 
marked  with  the  gross  weight  as  follows : 

Ib. 
Tare     100 

—  95 

—  83 

—  81 

359  total  tare. 

neat. 

2.  What  is  the  neat  weight  of  4  barrels  of  Indigo,  No,- 
and  weight  as  follows :      C.  qr.  Ib.  Ib. 

jVo.  1  —  4     1     10  Tare  36} 
o  _  3    3    02    —     39  ( 
3  _  4     0    19     —     32  f        cwt.gr.  Ib 
4_4    o      0    —     35J4^.1501I 

CASE  II. 

When  the  tare  is  at  so  much  per  box,  cask,  bag,  &c. 
Multiply  .the  tare  of  1  by  the  number  of  bags,  bales,  &c 
the  product  is  the  whole  tare,  which  subtract  from  the  gros? 
and  the  remainder  will  be  the  neat  weight. 

EXAMPLES. 

1.  In  4  hhds.  of  sugar,  each  weighing  10  cwt.  1  qr.  15  Ib 
gross;  tare  75  Ib.  per  hhd.  how  much  neatl 

Cwt.  qrs.  Ibs. 

10     115  gross  weight  of  one  hhd. 

4  [Carried  up.l 


C. 

qr. 

Ib. 

No.  1  —  9 

0 

12 

2  —  8 

3 

4 

3  —  7 

1 

0 

4  —  6 

3 

25 

Whole  gross  32 

0 

13 

Tare  359  lb.=  3 

3 

23 

Ans.  28 

3 

18 

TARE  AXD  TUKT.  105 


41     2      4  gross  weight  of  the  whole: 
~>5x4=2    2    20  whole  tare. 


8     3    12  neat. 

2.  What  is  the  neat  weight  of  7  tierces  of  ricre,  each 
^veiirhinjr  4  cwt.  1  qr.  9  Ib.  gross,  tare  per  tierce  34  Ib.  ? 

Ans.  28  C.  0  qr.  21  Ib. 

3.  In  9  firkins  of  butter,  each  weighing  2  qrs.  12  Ib.  gross<, 
tare  1  lib.  per  firkin,  how  much  neat?    Ans.  4  C.  2  qrs.  $  Ib. 

4.  If  241  bis.  of  figs,  each  3  qrs.  19  Ib.  gross,  tare  10  Ib, 
per  barrel  ;  how  many  pounds  neat  1  Ans.  22413. 

5.  In  16  bags  of  pepper,  each  85  Ib.  4  oz.  gross,  tare  per 
bag,  3  Ib.  5  oz.  ;  how  many  pounds  neat  ?        Ans.  1311. 

0.  In  75  barrels  of  figs,  each  2  qrs.  27  Ib.  gross,  tare  in  the 
whole  597  Ib.  ;  how  much  neat  weight  ?    Ans.  50  C.  1  qr. 

7.  What  is  the  neat  weight  of  15  hhds.  of  Tobacco,  each 
wfiighinjr  7  cwt.  1  qr.  13  Ib.  tare  100  Ib.  per  hhd.  ? 

Ans.  97  C.  (tor.  11  Ib. 
CASE  III. 

When  the  tare  is  at  so  much  per  ewt.  —  Divide  the  gross 
weight  by  the  aliquot  part  of  a  cwt.  for  the  tare,  which  sub- 
tract from  the  gross,  and  the  remainder  will  be  neat  weight. 

EXAMPLES. 

1.  What  is  the  neat  weight  of  44  cwt.  3  qrs.  16  Ib.  grosa, 
tare  14  Ib.  per  cwt.?     C.  qrs.  Ib. 

[  141b.  |  |  ]  44    3     16     gross. 

5     2     12£  tare. 
Ans.  39     1       3J  neat. 

2.  What  is  the  neat  weight  of  9  hhds.  of  Tobacco,  each 
weighing  gross  8  cwt.  3  qrs.  14  Ib.  tare  16  Ib.  per  cwt.  ? 

Ans.  68  C.I  qr.  24  Ib. 

3.  What  is  the  neat  weight  of  7  bis.  of  potash,  each  weighing 
201  Ib.  gross,  tare  10  Ib.  pel-  cwt.  ?  Ans.  1281  Ib.  6  oz. 

4.  In  25  bis.  of  figs,  each  2  cwt.  1  qr.  gross,  tare  per.  cwt. 
16  Ib.  ;  how  much  neat  weight?  Ans.  48  cwt.  24  Ib. 

5.  In  83  cwt.  3  qrs.  gross,  tare  20  Ib.  per  cwt.  what  neat 
weight?  Ans.  68  cwt.  3  qrs.  5  Ib. 

6.  In  45  cwt.  3  qrs.  21  Ib.  gross,  tare  8  Ib.  per  cwt,  how 
flinch  neat  weight  ?  Ans.  42  cwt.  2  qrs.  17£  Ib. 

".'.  AVhnt  is  fire  vaTiro  of  tire  n-rnt  weight  of  8  hhds.  of  FU-* 


10t>  TARE  AND  TRET. 

gar,  at  $9,  54  cts.  per  cwt.  each  weighing  10  cwt.  1  qr.  14  Ib. 
gross,  tare  14  Ib.  per  cwt.  Ans.  $692,  84  cts.  2£w. 

CASE  IV. 
When  Tret  is  allowed  with  the  Tare. 

1.  Find  the  tare,  which  subtract  from  the  gross,  and  call 
the  remainder  suttle. 

2.  Divide  the  suttle  by  26,  and  the  quotient  will  bo  the 
tret,  which  subtract  from  the  suttle,  and  the  remainder  will 
IA>  the  neat  weight. 

EXAMPLES. 

1.  In  a  hogshead  of  sugar,  weighing  10  cwt.  1  qr.  12  Ib. 
gross,  tare  14  Ib.  per  cwt.,  tret  4  Ib.  per  104  Ib.,*  how  much 
neat  weight  1  Or  thus, 

cwt.    qr.     Ib.  cwt.     qr.     Ib. 

10     1     12  14=^=J)10     1     12  gross. 

4  115  tare. 

41  26)9     0       7  suttle, 

28  I 11^  tret. 

330  Ans.  8~~2~~24  neat. 

83_ 
14=1)1160  gross. 

145  tare. 
26)10l5  suttle. 

39  tret. 
Ans.  976  Ib.  neat. 

2.  In  9  cwt.  2  qrs.  17  Ib.  gro^,  tare  41  Ib.,  tret  4  Ib.  per 
104  Ib.,  how  much  neat  1  Ans.  8  cwt.  3  qrs.  20  Ib. 

3.  In  15  chests  of  sugar,  weighing  117  cwt.  21  Ib.  gross, 
tare  173  Ib.,  tret  4  Ib.  per  104,  how  many  cwt.  neat  ? 

Ans.  Ill  cwt.Zllb. 

4.  What  is  the  neat  weight  of  3  tierces  of  rice,  each  weigh- 
ing 4  cwt.  3  qrs.  14  Ib  gross,  tare  16  Ib.  per  cwt.,  and  allow- 
ing tret  as  usual  ?  Ans.  12  cwt.  0  qrs.  6  Ib. 

5.  In  25  bis.  of  figs,  each  84  Ib.  gross,  tare  12  Ib.  per  cwt. 
tret41b.  per  104  Ib. ;  how  many  pounds  neat?  Ans.  1803 -f- 

*  This  is  the  tret  allowed  in  London.  The  reason  of  divividing  by  2t>  ? 
because  4  Ib.  is  1-26  of  *04lb.  but  if  the  tret  is  at  any  other  rrtf,  other  r»a  " 
mu*t  he  tnkrr«  tucc.oTiiing  to  ftp)  rate  proposed,  &-<\ 


IAKIS  A:,D  TUE'I.  107 

0.  What  is  the  value  of  the  neat  weight  of  4  barrels  of 
Spanish  tobacco;  numbers,  weights,  and  allowances  as  fbl- 
ows,  at  9jd.  per  pound  1 

cwt.  qrs. 

No.  1  Gross  1 

l       Tret  4  Ib.  per  104  Ib. 

Jns.  £17  16^. 
_ 

CASE  V. 

When  Tare,  Tret,  and  ClolT,  arc  allowed : 
Deduct  the  tare  and  tret  as  before,  and  divide  the  suttlo 
>y  168  (because  2  Ib.  is  the  T-Jg-  of  3  cwt.)  the  quotient  will 
»e  the  clofi',  which  subtract  from  the  suttlc,  and  the  remain* 
r  will  be  the  neat  weight. 

EXAMPLES. 

1.  In  8  hogsheads  of  tobacco,  each  weighing  13  cwt.  3qre. 
31b.  gross,  tare  1071b.  per  hlid.,  tret  4  Ib.  per  104  Ib.,  and 

Ib.  per  3  cwt.,  as  usual ;  how  much  neat? 
cwtf .  grs.  Ib. 
13    3    23 
4 
55 
28 
443 

nan 

i"5^  Ib.  gross,  of  1  hhiU 

3 

4689  whole  gross. 
107X3=^321  tare. 

26)4368  suttle, 

168  tret. 
168)^00  suttle. 
25  clofF. 


Ans.  4175  neat  weight. 

&  What  is  the  neat  weight  of  26  cwt.  3  qrs.  20  ib.  gross, 
are  62  Ib.,  the  allowance  of  tret  and  cloff  as  usual  ? 

.  neat  25  ci&,  1  qr,  5Z£,  I  oz,  nearly  ;  omitting  fur- 
ther frac*  ?>??5.. 


103 


INTEREST. 
INTEREST  is  of  two  kinds;  Simple  and  Compound* 

SIMPLE  INTEREST. 

Simple  Interest  is  the  sum  paid  by  the  borrower  to  the 
lender  for  the  use  of  money  lent  ;  and  is  generally  ut  a  cer- 
tain rate  per  cent,  per  annum,  which  in  several  of  the  Uni- 
ted States  is  fixed  by  law  at  G  per  cent,  per  annum  ;  that  is, 
6/.  for  the  use  of  1007.  or  6  dollars  for  the  use  of  100  dol- 
lars for  one  year,  &c. 

Principal,  is  the  sum  lent. 

Rate,  is  the  sum  per  cent,  agreed  on. 

Amount,  is  the  principal  and  interest  added  together. 

CASE  I. 

To  find  the  interest  of  any  given  sum  for  one  year. 
RULE.  —  :Multiply  the  principal  by  the  rate  per  cent,  and  divide  tfio 
product  by  100  ;  the  quotient  will  be  the  answer. 

EXAMPLES. 

1.  What  is  the  interest  of  397.  11s.  Sid.  for  one  year  aj 
67.  per  cent,  per  annum  1 

£.  s.  d. 
39  11  8-!- 


2J37  10  3 
20 


7(50 


0|12  Ans.  £2  7s.  (k/.T{fe. 

2.  What  is  tlfc  interest  of  23Gi  10s.  4d-  for  a  year,  at  5 
per  cerft  ?  A  nr.  £  1 1 16*  &! . 


.  109 

3.  What  is  the  interest  of  571 1.  13s.  9d.  for*  one  year,  at 
G/.  per  cent.  ?  Ans.  £34  6s.  Q\d. 

4.  What  is  the  interest  of  2J.  12s.  9-J-d.  for  a  year,  at  67. 
per  cent.  1  Ans.  £0  3s.  %J. 

FEDERAL  MONEY. 

5.  What  is  the  interest  of  468  dols.  45  cts.  for  one  year, 
at  6  per  cent.  1  $     cts. 

468,  45 
6 


Ans.  28J10,  70=$28, 
Here  I  cut  off  the  two  right  hand  integers,  which  divide 
by  100  :  but  to  divide  federal  money  by  100,  you  need  only 
call  the  dollars  so  many  cents,  and  the  inferior  denomina- 
tions decimals  of*a  cent,  and  it  is  done. 

Therefore  you  may  multiply  the  principal  by  the  rate, 
and  place  the  separatrix  in  the  product,  as  in  multiplication 
of  federal  money,  and  all  the  figures  at  the  left  of  the  sepa- 
ratrix, will  be  the  interest  in  cents,  arid  the  first  figure  on 
the  right  will  be  mills,  and  the  others  decimals  of  a  mil!,  as 
in  the  following 

EXAMPLES. 

6.  Required  the  interest  of  135  dols.  25  cts.  for  a  year  at 
6  per  cent  ?  $  cts. 

135,   25 
6 


Ans.  811,  50=88,  11  cts.  3m. 

7.  What  is  the  interest  of  19  dols.  51  cts.  for  one  year,  at 
5  per  cent.  ?  §     cts. 

19,   51 


Ans.  97,    55=97  cts.  fym. 

8.  What  is  the  interest  of  436  dols.  for  one  year,  at  6  per 
cent.?  6 


Ans.  2616  cfe.=$26,  16  cte 


ANOTHER  METHOD. 

Write  down  the  given  principal  in  cents,  which  multiply 
by  the  rate,  and  divide  by  100  as  before,  and  you  will  have 
the  interest  for  a  year,  in  cents,  and  decimals  of  a  cent,  as 
follows : 

9.  What  is  the  interest  of  §73,  65  cents  for  a  year,  at  C 
per  cent.  ? 

Principal  7365  cent?. 
(j 

Ans.  441,90=441-^-  cts.  or  $4,  41  eta.  9m. 

10.  Required  the  interest  of  $85,  45  cts.  for  a  year,  at  7 
per  cent.  I 

Cents. 

Principal  8515 
7 


Jbi*.89S;  15  cents f=$o 
CASE    II 

To  find  the  simple  interest  of  any  sum  of  money,  for  any 
number  of  years,  and  parts  of  a  year. 

GENERAL  RULE. — 1st.  Find  the  interest  of  the  given  sum  for  one 
year. 

2d.  Multiply  the  interest  of  one  yearj>y  the  given  number  of  years, 
and  the  product  will  be  the  answer  for  that  time. 

3d.  If  there  be  parts  of  a  year,  us  months  and  days,  work  for  tho 
months  by  the  aliquot  parts  of  a  year,  and  for  the  days  by  the  Rule  of 
Three  Direct,  or  by  allowing  30  days  to  the  month,  and  taking  aliquot 
parts  of  the  same.* 


*  By  allowing  the  month  to  be  30  days,  and  taking  aliquot  parts 
you  will  have  the  interest  of  any  ordinary  sum  sufficiently  exact  for  common 
use  ;  but  if  thr  sum  be  very  large,  you  may  say, 

As  365  days  :  is  to  the  "interest  of  one  year  :  :  so  is  the  given  number  of 
days  :  to  the  interest  required, 


.•SIMPLE  INTEREST. 


ill 


EXAMPLES. 

I.  What  is  the  interest  of  751.  8s.  4d.  for  5  years  and  2 
months,  at  61.  per  cent,  per  annum  ? 
£.     s.     d. 
75     8     4  £.  s.  d. 

6     2  w0.=£)4    10  6  Interest  for  1  year. 
5 


4152    10     0 
20 

10J50~ 
12 


22   126  do.  5  years. 
0   15  1  do.  for  two  months. 


6|00 


£23     7  7  Ans. 


2.  What  is  the  interest  of  64  dollars  58  cents  for  3  years, 
5  months,  and  10  days,  at  5  per  cent.  ? 
$  64,58 
5 


j  322,90     nterest  for   1  year   in  cents,  per 
3  [Case  I. 


968,70  do.  for  3  years. 
4  mo.       i     107,63  do.  for  4  months. 
1  mo.        V  j     26,90  do.  for  1  month. 
10  days,    ^  J       8,96  do.  for  10  days. 

Ans.  1112,19=1112cfe.  or  $  11,  12c.  l^m. 

3.  What  is  the  interest  of  789  dollars  for  2  years,  at  6 
per  cent.  1  Ans!  $94,  68  cts. 

4.  Of  37  dollars  50  cents  for  4  years,  at  6  per  cent,  per 
annum  ?  Ans.  900  cts.  or  $9. 

5.  Of  325  dollars  41  cts.  for  3  years  and  4  months,  at  5 
per  cent.  ?  Ans.  $54,  23  cts.  5  m. 

6.  Of  3257.  12s.  3d.  for  five  years,  at  6  per  cent.  ? 

Ans.  £97  135.  Sd. 

7.  Of  174/.  10s.  6d.  for  3  and  a  half  years,  at  6  per  cent.? 

Ans.  £36  13*. 

8.  Of  150/.  16s.  8d.  for  4  vears  and  7  months,  at  6  per 
?  Aw.  £4!9*.7ff 


1J2  COMMISSION. 

9.  Of  1  dollar  for  12  years,  at  5  per  cent.? 

Ans.  60  cte. 

10.  Of  215  dollars  34  cts.  for  4  arid  a  half  years,  at  3 
and  a  half  per  cent.  Ans.  $33,  91  cts.  6m. 

11.  What  is  the  amount  of  3*24  dollars  61  cents  for  5 
years  and  5  months,  at  6  per  cent.  ? 

Ans.  $430,  10  cts.  S^m. 

12.  What  will  30007.  amount  to  in   12  years  and  10 
months,  at  6  per  cent.  ?  Ans.  £5310. 

13.  What  is  the  interest  of  2577.  5s.  Id.    for  1  year  and 
3  quarters,  at  4  per  cent.  ?  Ans.  £18  Os.  Id.  3qrs. 

14.  What  is  the  interest  of  279  dollars   87  cents  for  2 
vears  and  a  half,  at  7  per  cent,  per  annum  ? 

Ans.  $48,  97c*s.  1\m. 

15.  What  will  279Z.  13s.  8d.  amount  to  in  3  years  and  a 
half,  at  5£  per  cent,  per  annum? 

Ans.  £331  Is.  6d. 

16.  What  is  the  amount  of  341  dols.  60  cts.  for  5  years 
and  3  quarters,  at  7  and  a  half  per  cent,  per  annum  ? 

Ans.  $488,  9l£  cts. 

17.  What  will  730  dols.  amount  to  at  6  per  cent,  in  5 
years,  7  months,  and  12  days,  or  ---/j  of  a  year  ? 

Ans.  $975,  99  cts. 

18.  What  is  the  interest  of  1825Z.  at  5  per  cent,  per  an- 
num, from  March  4th,  1796,  to  March  29th,  1799,  (allow- 
ing the  year  to  contain  365  days  ?) 

Ans.  £280. 

NOTE. — The  Rules  for  Simple  Interest  serve  also  to  cal- 
culate Commission,  Brokerage,  Ensurance,  or  any  thing 
•else  estimated  at  a  rate  per  cent. 

COMMISSION, 

IS  an  allowance  of  so  much  per  cent.,  to  a  factor  or  cor- 
respondent abroad,  for  buying  and  selling  goods  for  his  em- 
ployer. 

EXAMPLES. 

1.  What  will  the  commission  of  843/.  10s,  come  to  at  5 
per  rf^nt.  ? 


EAGE. 


£.     f.  Or  thus, 

843  10  £•      *• 

5  £5  is  aV)843     10 


42|  17  10                                         -4ns.  £42  3    6 
20  


3150 
12 

6[00  £42  3*.  6cL 

2.  Required  the  commission  on  964  dols.   90  cts.  at  2} 
per  cent.  1  Ans.  $21 ,  71  cts. 

3.  What  may  a  factor  demand  on  1^  per  cent,  commis-* 
sion  for  laying  out  3568  dollars  1  Ans.  §62,  44c?s. 

BROKERAGE, 

IS  an  allowance  of  so  much  per  cent,  to  persons  assist- 
ing merchants,  or  factors,  in  purchasing  or  selling  goods. 

EXAMPLES. 

1.  What  is  the  brokerage  of  750J.  8s.  4d.  at  6s.  8d.  pel- 
cent.  ? 

£     s.  d. 

750  8  4     Here  I  first  find  the  brokerage  at  1  pound 
I          per  cent,  and  then   for  the  given  rate* 

which  is  £  of  a  pound. 
7,50  8  4 

20    ;  6-.  d.       £.  s.  d.  qrs. 

6  8=i)7  10  1    0 
10,08 

12  Ans.  £2  10  0    1-J- 


1,00 

2.  What  is  the  brokerage  upon  4125  dols.  at  £  or  75  cents 
per  cent.  1  Ans.  §30,  93  cts.  7£  m. 

3.  If  a  broker  sell  goods  to  the  amount  of  5000  dollars, 
what  is  his  demand  at  65  cts.  per  cent.  1 

Ans.  §32.  50  ct*. 
K  rt 


:yl4   •  X,          L 

4.  What  may  a  broker  demand,  when  he  sells  goods  to 
the  value  of  508/.  17s.  lOd.  and  I  allow  him  U  per  cent.  ? 

Ans.  £1  12s.  Sd. 


ENSURANCE, 

IS  a  premium  at  so  much  per  cent,  allowed  to  persons 
and  offices,  for  making  good  the  loss  of  ships,  houses,  mer- 
chandise, &c.  which  may  happen  from  storms,  fire,  <kc. 

EXAMPLES. 

1.  What  is  the  ensurance  of  7257.  8s.    lOd.  at  12  J  per 
cent.?  Ans.  £90  13s.  7±d. 

2.  What  is  the  ensurance  of  an  East-India  ship  and  car- 
go, valued  at  123425  dollars,  at  15*  per  cent.  1 

Ans.  $19130,  87  cts.  5m. 

3.  A  man's  house  estimated  at  3500  dols.,  was  ensured 
against  fire,  for  If  per  cent,  a  year :   what  ensurance  did 
he  annually  pay?  Ans.  $61,  25 


Short  Practical  Rules  for  calculating  Interest  at  6  per  cent, 
either  for  months,  or  months  and  days. 

1.  FOR  STERLING  MONEY. 

RULE. — 1.  If  the  principal  consist  of  pounds  only,  cut  off  the  unit 
figure,  and  as  it  then  stands  it  will  be  the  interest  for  one  month,  in 
shillings  and  decimal  parts. 

2.  If  the  principal  consist  of  pounds,  shillings,  &c.  reduce  it  to  its 
decimal  value  ;  then  remove  the  decimal  point  one  place,  or  figure, 
flirther  towards  the  left  hand,  and  as  the  decimal  then  stands,  it  will 
show  the  interest  for  one  month  in  shillings  and  decimals  of  a  shil- 
ling. 

EXAMPLES. 

1.  Required  the  interest  of  54  J.  for  seve,n  months  and  ten 
days,  at  6  per  cent. 


SHORT  PRACTICAL  RULES, 
S. 

10  days=i)5,4  Interest  for  one  month. 

7 

37,8  ditto  for  7  months. 
1,8  ditto  for  10  days. 

Ans.  39,6  shillings=£l  195. 
12 


2.  What  is  the  interest  of  421.  10s.  for  11  months,  at  6 
per  cent.  1 

£.     s.          £. 

42    10  =  42,5  decimal  value. 

Therefore  4,25  shillings  interest  for  1  month. 

_Ji  *.  ,   a. 

Ans.     46,75    Interest  for  11  mo.  =>  2      6      9 

3.  Required  the  interest  of  94?.  7s.  6d.   for  one  year, 
five  months  and  a  half,  at  6  per  cent,  per  annum  ? 

Ans.  £8  55.  Id.  3,5grs. 


4.  What  is  the  interest  of  12/.  18s.  for  one  third  of  a 
month,  at  6  per  cent.  ? 


H.  FOR  FEDERAL  MONEY. 

RULE.— 1.  Divide  the  principal  by  2,  placing  the  separatrix  as  usual, 
ami  the  quotient  will  be  the  interest  for  one  month  in  cents,  and  deci- 
mals of  a  cent ;  that  is,  the  figures  at  the  left  of  the  separatrix  will 
cents,  and  those  on  the  right,  decimals  of  a  cent. 

2.  Multiply  the  interest  of  on«  month  by  the  given  number  of 
months,  or  months  and  decimal  parts  thereof,  or  for  tho  days  take  the 
even  parts  of  a  momth.  &c. 


1 1(3  SHORT  I'RACTICL  RULES 

EXAMPLES. 

1.  What  is  the  interest  of  341  dols.  52  cts.  for  7£  months  ¥ 
2)341,52 

Or  thus,  170,76  Int.  for  1  month. 
170,76  Int.  for  1  month.          X  7,5  months. 


85380 


1 195,32  do.  for  7  mo.  1 19532 

85,38  do.  for  ^  mo.  $  cts.  m. 

1280,700cfc.  =  12,80  7 

1280,70  Ans.     12SO,7c>s.-=$12,  SQcts.  7m. 
2.  Required  the   interest  of  10  dols.  44  cts.  for  3  years, 
5  months,  and  10  days. 
2)}0,44 

10  days— £)     5,22  interest  for  1  month. 
41  months. 


5,22 

208,8 

214,02  ditto  for  41  months. 
1,74  ditto  for  10  days. 


215,76  cts.    Ans.  =$2,  15  cts.  7  m.+ 
3.  What  is  the  interest  of  342  dollars  for  11  months'? 

The  £  is  171  interest  for  one  month. 
11 

Ans.  1881  cte.=$18,  81  cts. 

NOTE. — To  find  the  interest  of  any  sum  for  two  months, 
at  6  per  cent,  you  need  only  call  the  dollars  so  many  cents, 
and  the  inferior  denominations  decimals  of  a  cent,  arid  it  is 
done :  Thus,  the  interest  of  100  dollars  for  two  months,  is 
100  cents,  or  one  dollar ;  and  $25,  40  cts.  is  25  cts.  4  m. 
&c.  which  gives  the  following 

RULE  II. — Multiply  the  principal  by  half  the  number  of  months, 
and  the  product  will  show  the  interest  of  the  given  time*  in  cent*  and 
<!e<:iroats  of  a  rent,  as  above. 


1.  Required  the  interest  of  316  "dollars  for  1  year  and  10 
months.  11=£  the  number  of  mo. 

Am.  3476  cts.  =$34,  76  cts. 

2*  What  is  the  interest  of  364  dols.  25  cts.  for  4  months  1 

§     cts. 
364,  25 

2  half  the  months. 


728,  50  cts.     Ans.=$7,  28  c£s.  5  m. 

III.  When  the  principal  is  given  in  federal  money,  at  6 
per  cent,  to  find  how  much  the  monthly  interest  will  be  in 
New-England,  &c.  currency. 

RULE.  —  Multiply  the  given  principal  by  ,03,  and  the  product  will  be 
the  interest  for  one  month,  in  shillings  and  decimal  parts  of  a  shilling. 

EXAMPLES. 

1.  What  is  the  interest  of  325  dols.  for  11  months  ? 


9,75  shil.  int.  for  one  month 
X  11  months. 


-4ns.  107,25  s.=£5  7*.  3d. 

1.  What  is  the  interest  in  New-England  currency  of  31 
dols.  68  cts.  for  5  months  1 

Principal  31,68  dols. 
,03 

,9504  Interest  for  one  month. 
5 


Ans.  4,7520*.=4s.  M. 
12 


118  SHORT  PRACTICAL  RULES 

W.  \\yien  $lfe  principal  js  fri^tetf  in.  pmmds,  Shillings,  &c. 
ev\8^I£n^£ntl  curjpncy-at  (juner  ce£j£tp  find  ho$y  much  the 
monthly  interest  will  be  in  Sacral  money. 

RULE. — Multiply  the  pounds,  &c.  by  5,  and  divide  that  product  by 
3,  the  quotient  will  be  the  interest  for  one  month,  in  cents,  and  deci- 
mals of  a  cent,  &c. 


EXAMPLES. 


1.  A  note  for  £411  New-England  currency  has  been  on 
interest  one  month  ;  how  much  is  the  interest  thereof  in  fe- 
deral money  ?  £. 
411 


3)2055 


s.=--$6,  S5cts. 

2.  Required  the  interest  of  39/.  18s.  N.  E.  currency,  for 
7  months  ?  £ 

39,9  decimal  value. 
5 


3)199,5 

Interest  for  1  mo.  66,5  cents. 

7 


Ditto  for  7  mo.   465,5  cfc.=$4,  65  cts.  5  m.  Ans. 

* 

V.  When  the  principal  is  given  in  New-England  and  Vir- 
ginia currency,  at  6  per  cent,  to  find  the  interest  for  a  year, 
in  dollars,  cents,  and  mills,  by  inspection. 

RULE. — Since  the  interest  of  a  year  will  be  just  so  many  cents  as 
the  given  principal  contains  shillings,  therefore,  write  down  the  shil- 
lings and  call  them  cents,  and  tho  pence  in  the  principal  made  less  by 
1  it  they  exceed  3,  or  by  2  when  they  exceed  9.  will  be  the  milte,  very 


00.  CALCULATING  ISTTEEEST  .  1  1  9 

EXAMPLES. 

1.  What  is  the  interest  of  2/.  5s.  for  a  year,  at  G  per  ct.  ? 

«£2  55=455.    Interest  45  cts.  the  Answer. 

2.  Required  the  interest  of  100/.  for  a  year,  at  6  per  ct.  ? 

£100=20005.    Interest  2000  c*5.=:$20  Ans. 

3.  Of  27s.  6d.  for  a  year? 

Ans.  27s.  25  27  cts.  and  Gd.  is  5  m. 

4.  Required  the  interest  of  51  10s.  lid.  for  a  year  ? 

£5  105.=1105.    Interest  110cfe.=$l,  Wets.  Ow. 
1  1  pence.  —  2  per  rule  leaves  9—  9 

Ans.$l,  10       9 

VI.  To  compute  the  interest  on  any  note  or  obligation,, 
when  there  are  payments  in  part,  or  endorsements. 

RULE.  —  1.  Find  the  amount  of  the  whole  principal  for  the  whole 
time. 

2.  Cast  the  interest  on  the  several  payments,  from  the  time  they 
M'ere  paid,  to  the  time  of  settlement,  and  rind  their  amount  ;  and  lastly, 
deduct  the  amount  of  the  several  payments  from  the  amount  of  tho 
principal. 

EXAMPLES. 


Suppose  a  bond  or  note  dated  April..  17,  1793,  was  pve 
for  675  dollars,  interest  at  (>  per  cent,  and  there  were  pay- 
ments endorsed  upon  it  as  follows,  viz. 
First  payment,  148  dollars,  May  7,  1794. 
Second  payment,  341  dols.  August  17,  179(5. 
Third  payment,  99  dols.  Jan.  2,  1798.     I  demand  how 
much  remains  due  on  said  note,  the  17th  June,  1798  ? 

$     cts. 

148,  00  first  payment,  May  7,  1794.  Yr.  mo. 

36,  50  interest  up  to  —  June  17,  1798.=4     1£ 

184,  50  amount 


341,  00  second  payment,  Aug.  17,  1796.  Yr.  mo. 
37,  51  interest  to June  17,   1798.  —1     10 

378,  51  amount. 

"vied  over."j 


SHORT  raACriCAL 

$    cts. 

99,  00  third  payment,  January  2,  17SB. 
2,  72  interest  to — June  17,  1798.=  5J 


101,  72  amount. 
184,  50 

several  amounts. 


184,  50  J 
378,  51  > 
101,  72) 


664,  73  total  amount  of  payments. 

675,  00  note,  dated  April  17,  1793.          Yr.  mo. 
209,  25  interest  to—  June  17,  1798.       —5    2 


884,  25  amount  of  the  note. 
6154,  73  amount  of  payments. 

$219,  52  remains  due  on  the  note,  June  17,  1798. 
2.  On  the  16th  January,  1795, 1  lent  James  Paywell  500 
dollars,  on  interest  at  6  per  cent,  which  I  received  back  in 
the  following  partial  payments,  as  under,  viz. 

1st  of  April,  1796  *  -       $  50 

16th  of  July,  1797  -    400 

1st  of  Sept'.  1798  60 

How  stands  the  balance  between  us,  on  the  16th  Novem- 
ber, 1800  1  Ans.  due  to  me,  $63,  18  cts. 

3.  A  PROMISSORY  NOTE,  VIZ. 

£62105.  New-London,  April  4,  1797. 

On  demand,  I  promise  to  pay  Timothy  Careful,  sixty-two 
pounds,  ten  shillings,  and  interest  at  6  per  cent,  per  annum, 
till  paid ;  value  received. 

JOHN  STANBY,  PETER  PAYWELL, 

RICHARD  TESTIS. 

Endorsements.  £.   s. 

1st.  Received  in  part  of  the  above  note,  . 

September  4,  1799,  50    0 

And  payment  June  4, 1800,  12  10 

How  much  remains  due  on  said  note,  the  4th  day  of  De- 
cember, 180CX  £.  s.  d. 

4ns.  9  12  G 


FoR  oALuL-LATKvu  IXTERJjST.  1:21 

%  .      f 

NOTEU — T^hfi  preceding  Ruler  by-  c"us.to.th;,  is  r$ hcferefl  so* 
p*qpular»  and  sx>  much:  practised  and  esteemed  by*  irra'ny  QA 
account  of  its  being  simple  and  concise,  that  I  have- given  it 
a  place  :  it  may  answer  for  short  periods  of  time,  hut  in. 
a  long  course  of  years.,  it  will  be  found  to-  be  very  errone- 
ous. 

Although  this  method  seems  at  first  view  to  be  upon  the 
ground  of  simple  interest,  yet  upon  a  little  attention  tire 
following  objection  will  be  found  most  clearly  to  lie  against 
it«  viz.  that  the  interest  will,  in  a  course  of  years,  complete- 
ly expunge,  or  as  it  may  be  said,  eat  up  the  de.hr.  For  an 
explanation  of  this^  take  the  following 


A  lends  B  lOi)  dollars-,  at  C  per  c£ht.  interest,  and  takes 
his  note  of  hand  ;  B  does  no  more  tlraa  pay  A  at  eX«ery 
};eitr*s  end  6  dollars,  (which  is  then  Justly1  due1  toi  B  -fop  tlie 
use  6f  his  money)  and  has  it  endorsed  an  his  no£ev  At  the 
and  of  10  year-s  B  takes  up  his  note,  and  the  sum  he  lias  to 
jtny  is  reckoned  thus :  The  principal  100  dollars,  art  itite- 
rei  10  year's  amounts  to  160  dollars  ;  there  are  nine*  en- 
dorjaements  of  G  dollars  ench,  upon  which  the  debtor  claims 
fteresj: ;  one  for  nine  years,  the  second  for  S  years,  the 
third  for  7  years,  and  so  down  to  the  time  of  settletnejtt^ 
the  whole  amount  of  the  several  endorsements  and  theifin- 
tei*est,  (as  any  one  can  stse  by  casting  it)  is  $70, 20  cts.  thi<? 
subtracted  from  160  dols.  the  amount  of  the  debt,  leaves  inr 
favour  of  the  creditor,  $80,  40  cts.  or  $10,  20  cts.  les^  than 
tire  original  principal,  of  which  he  has  nol  received  a  cerr^ 
hul  only  its  annual  interest* 

If  the  same  note  should  lie  2D  years  in  the  same*  way,  B 
w:ojuld  owe  but  &7  dols.  6.0  cts.  without  paying  tlie  least 
fraction  of  the  100  dollars  borrowed. 

Extend  it  to  28  years,  and  A  the  creditor  would  fall  i/i 

cjfcbt  to  B,  without  receiving  a  cent  of  the  100  dols.  which 

"fie  lent  him.     Se£  a  hotter  Rnle  in  Simple  Interest  by  de- 
cimals, pnge"  lY-5. 

7 


IS  when  tlie  interest  is  added  to  tile  principal  at  the 
£fld  of  the  year,  and  on  that  amont  the  interest  cast  for  ano- 
ther year,  and  added  again,  and  s,o  on  :  this  is  called  inte- 
rest upon  interest. 

RULE.  —  Find  the  interest  fo.r  a  year,  a'nd  add  it  to  th&  princi- 
pal, which  call  the  amount  for  the  first  year  ;  find  the  interest 
pP^this  amount,  which  ad'd  as  before,  for  the  amount  of  the  se- 
cond, and  so  on  for  any  number  of  years  required.  Subtract  the 
original  principal  from  the  last  amount,  and  the  remainder  will  be 
tfie-Compoujid  Interest  for  the  whole  time. 


1.  Required  the  amount  of  100  dollars  for  3  years  'at  6 
per  cent,  per  annum,  compound  interest  ? 

$  cts.  $  cts. 

1st  Principal  100,00  Atruount  106,00  for  1  yei\r, 
3d  Principal  106,00  Amount  ]  12,36  for  2  years. 
3d  Principal  113)36  Amount  119,1016  for  3  yrs.  Arts' 

2.  What  is  the  amount  of  425  dollars,  for  4  years,  at  5 
per  cent,  per  annum,  compound  interest? 

Ans.  #516,  59  cts. 

3.  What  will  400J.  amount  to,  in  four  years-,  at  6  per 
cent,  per  annum,  compound  interest? 

A»s.  £504  19*.  9^ 

4.  What  is  the  compound  interest  of  150?.   10s.   for  3 
years,  at  6  per  cent,  per  annum?    Ans.  £28  14s.  ll£d.-f 

5.  What  is  the  compound  interest  of  500  dollars  ft>r  4 
years,  at  6  per  cent,  per  annum  ?  Ans.  $131,238+ 

6.  What  will  1000  dollars  amount  to  in  4  years,  at  7  per 
cent,  per  annum,  compound  interest? 

Ans.  $1310,  79  cfe»6.m»  -f 

7.  What  is  tile  amount  of  750  dollars  for  4  years-,  at  6 
per  cent,  per  annum,  compound  interest? 

Ans.  1946,  85  ct$.  7,72  #h 

8.  What  is  the  compound  interest  of  876  dols.  90  cents 
for'3J-  years,  at  6  per  cent,  per  annum? 


t  BS.  •  V-:.;  {'& 

EXAMPLES. 

1.  It*  an  annuity  of  707.  be  forborne  o  y.ears,  what  will 
be  due  for  the  principal  and  interest  at  the  end  of  said 
term,  simple  interest  being  computed  at  5  per  cent,  per 
annum?  Yr.  £.  s. 

^Hfr-  Interest  of  70/.  at  5  per  ce'n't.  for          1 —  3     10 
at  the  giveji  rau*  2 —  7       0 

deb*.  3—10     LO 

4 14       Q 

RULE.— As  the  amount  of  1007.  or  1.00  dollars,  at  ..~_0        ft 
ami  time :  is  to  the  interest  of  100,  at  the  sam'e  rjate  and  time  :  . :? 
tfte  given  sum  :  to  the  discount- 
Subtract  the  discount  from  the  given  sum.,  and  {lie -remainder  is  the 
present  worth. 

Or — as  the  amount  of  100  :  is  to.  IjOO  :  :  so  is  tluj  given  sum  or 
debt  :  to  the  present  worth. 

PROOF. — Find  the  amount  of  the  present  worth,  at  tire  given 
rate  and  time,  and  if  the  work  is  right,  tfra't  will  be  equal  t£  the 
^fiven  sum. 

KXA&PLES. 

1..  What  must  be  discounted  for  the  fendy  payment  of 
100  d'attars,  due  a  year  hence  at  6  per  cent,  a  year  ? 

$       S  $        §  cts. 

Afe  1QG  :  6  :  :  100  :  5  66  the  answer, 
100,00  given  sum* 
5jG6  discount. 

$94,34  the  present  worth*. 

&  "What  sum  in  ready  money  \vill  discharge  a  debi  of 
.  due  1  year  and  8  months  b<5npe4  at  6  TKT  cent.  ? 
£100      * 

10  interest  for  20  months.. 

110  Am't  ^.        £..          £.        £v    s.  <l 

As  110  :  100  :  :  925  :  840  18  2+ An*, 
3.  What  is  the  present  worth  of  600  dollars.,  dire  4  years 
hfcnce,  at  5  per  cent.  1  Ans.  £500, 

4-  What  is  the  discount  o/  2757.  10».  far  10  months,,  at 
6  per  ceftn  .pn:  anmim  ?  An*.  ' 


IS  when  tlie  interest  is  added  to  tile  pmicipaK  at  the 
*mi  of  the  year,  and  on  that  amont  the  interest  cast  for  ano- 
ther year,  and  added  again,  and  $o  on  :  this  is  called  inte- 
rest upon  interest. 

>    ,n+ 

RULE.  —  FTnd   the  interest  TQF  a  year    *-T-<?  paid   at  diflfereift 
pah  which  call  the  amount  fo/csent  worth  of  each  particular 
^^Sy:  and  when  so  found,  add  them  intp  on* 


7.  What  is  the  discount  of  756/.  the  one  half  payable  iu 
six  months,  and  the  other  half  in  six  months  after  that,  at  7 
per  cent.  1  sbis.  £37  10s.  2^7. 

8.  If  a  legacy  is  left  me  of  2000  dollars,  of  which  500 
rtols.  are  payahle  in  6  months,  800  dols.  payable  hi  I  year, 
and  the  rest  at  the  end  of  3  years  ;  how  much  ready  money 
ought  I  to  receive  for  said  legacy,  allowing  6  per  cent,  dis- 
count 1  Arts.  |1833,  37  cte.  4  m. 


ANNUITIES. 

AN  Annuity  is  a  sum  of  money,  payable  every  year,  or 
for  a  certain  number  of  years,  or  for  ever. 

"When  the  debtor  keeps  the  annuity  in  his  own  hamTs 
Beyond  the  time  of  payment,  it  is  said  to  be  in  arrears. 

The  §um  of  all  the  annuities  for  the  time  they  have  been 
forebarne,  together  with  the  interest  due  on  each,  is  callfed 
the  amount. 

If  an  annuity  is  bought  off,  or  paid  all  at  once  at  tbe 
beginning  of  the  first  year,  the  price  which  is  paid  for  ij:  is 
called  the  present  worth. 

To  find  the  Amount  of  an  annuity  at  simple  interest: 
RifLE.— 1.  Find  the  interest  of  the  given  annuity  for  t  year. 

2.  And  then  for  2,  3,  &c.  yrars,  up  to  the  given  time,  less  1. 

3.  Multiply  the  annuity  by  the  number  of  ytjars  given,  anti  axlti 
the  product  to  the  \vhV)!e  interest,  and  the  sum  will   he  t&e 
sought, 


,  ES,  •  :•:.;  -l-A 

EXAMPLES. 

1.  It*  an  annuity  of  707.  be  forborne  o  \;ears,  what  will 
be  due  for  the  principal  and  interest  at  the  end  of  said 
term,  simple  interest  being  computed  at  5  per  cent,  per 
annum?  Yr.  £.  s. 

isfc  Interest  of  707.  at  5  per  cent,  for          1 —  3     10 

2—7       0 

3—10     10 

4—14       0 

2<k  And  5  71*3.  annuity,  at  707.  per  yr.  is        350      0 

Ans.  £385       0 

21.  A  house  being  let  upon  a  lease  of  7  years,  at  400 
dollars  per  annum,  and  the  rent  being  in  arrear  for  the' 
whole  term,  I  demand  the  sum  due  at  the  end  of  the  term, 
shnpj'e  interest  being  allowed  at  6?.  per  cent,  per  annunf? 

Ans.  £330'4. 


To  find  the  present  toortji  of  an  annuity  at  simpie 
interest. 


.  —  Find  the  present  worth  of  each  year  by  itself,  discounting 
from  the  time  it  falls  duo,  and  the  sum  of  all  these  present*  Wjorths 
v\;ill  be  the  present  worth  required. 

EXAMPLES. 

1.  What  is  the  present  worth  of  400  dols.   per  annum. 
to  continue  4  years;,  at  6  per  cent,  per  annum  1 
106}  377,35849  =  Pres.  worth   of  1st  yr. 

H2  (  .  10()  .  .  jm  .  357,14285  -      -,  -  -  2d  yr. 

118  f  '  1(          4UD  '  338,98305  «      -  -- 
1243  322,58064-:      —  -  4th  vf. 


Ans.  $1396,06o03  =  $139G,  Get*  5m. 
How  ni/rch  present  money  is  equivalent  to  an  annuity 
of  100  dollars,  to  continue  3  years ;    rebate  being  raaidfe  at 
6  per  cent,  t  Ans.  $2'68,  37c^.  lm. 

3.  What  is  807*  yearly  rent,  to  continue  5  years,  worth 
in  ready  rotfnfcy,  a*t  6?.  per  cent.  ?          Ans.  £54O  15s.-f 


EQUATION  OF  PAYMENTS, 

IS  finding  the  equated  time  to  pay  at  once,  several  debts 
clue  at  diiferent  periods  of  time,  so  that  no  loss  shall  »be 
sustained  by  either  party. 

RULE. — Multiply  each  payment  by  its  time,  and  divide  the  suni-of 
the  several  products  by  the  whole  debtt  emu  the  quotient  will  be  the 
equated  time  for  the  payment  of  the  whole. 

EXAMPLES. 

1.  A  owes  B  380  dollars,  to  be  paid  as  follows5 — viz.  100 
dollars  in  6  months,  120  dollars  in  7  months,  and  160  dol- 
lars in  10  months  :    What  is  the  equated  time  for  the  pn\> 
inent  of  trre  whole  debt  ? 

100  x  6  =  600 
120  x  7-840 
160  x  10  =  1GOO 

380  }3040(8  months;    An*. 

2.  A  merchant  hath  owing  him  800J.  to  be  paid  as  &!- 
laws  :  507.  at  2  months,  100/.  at  5  months,  and  the  rest  at 
8  months ;  and  it  is  agreed  to  make  one  payment   of  the 
vvtiole  :  I  demand  the  equated  time  ?         Ans.  6  months-. 

3.  P  owBsH  1000  dollars,  whereof  200  dollars  is  to  be 
jpaid  present,  400  dollars  at  5  months,  and  the  rest  at  1*5 
months,  but  they  agree  to  make  one  payment  of  the  whole; 
I  demand  when  that  time  must  be  1          Ans.  S  months-. 

4.  A  i&erclaant  has  due  to  him  a  certain  sum  ofmorrey, 
to  he  paid  one  sixth  at  %  months,  one  third  at  3  months, 
and  the  rest  at  6  months  ;  "what  is  the  equated  time  for  the 
jgrayment  of  the  whole  ?  Ails.  4|  months. 

BARTER, 

JS  the  exfchaaging  of  one  commodity  fop  another,  and 
iftrjeets  merchants  and  traders  how  to  make  tUe  exfchaorge 
without  loss  to  either  party. 

RtiLE. — Find  the  value  of  the  Commodity  whose  tfuattUty  is  given  ; 
then  find  what  quantity  of  the  fcthep  at  the  g^posed  -rate  can,  be 
bought  for  the  same  money,  and  it  gives  the,answer>. 


127 


EXAMPLES, 


1.  \Vl*at  quantity  of  flax  at  9  cts.  per  Ib.  must  be  giveji 
in  barter  for  12  Ib.  of  indigo,  at  2  dols.  19  cents  per  Ib.  ? 

12  Ib.  of  indigo  at  2  dols.  19  Cts.  per  Ib.  comes  to  28 
rfols.  28  cts.— therefore,  As  9  cts.  :  1  Ib.  :  :  26,28  cts.  '. 
292  the  answer. 

2.  How  much  wheat  at  1  dol.  25  cts.  a  bushel,  must  he 
given  in  barter  for  50  bushels  of  lye,  at  70  cts.  a  bushel  1 

Ans.  28  bushels. 

3.  How  much  rice  at  28s.  per  cwt.  must  be  bartered  fq'r 
<U  <xwt.  of  raisins,  at  5d.  per  Ib.  t 

Ans.  5  cwt.  3grs.  9J-J§». 

4.  How  much  tea  at  4s.  9d.  per  Ib.    must   be    given    in 
flatter  for  78  gallons  of  brandy,  at  12s.  3-Jd.  per  gallon  ? 

Ans.  201  Ib.  13tfoz. 

o".  A  arid  B  bartered  :  A  had  B\  cwt.  of  sugar  at  12  els. 
pe*  Ib.  for  which  B  gave  him  18  cwt.  of  flour ;  what  \ras 
the  flour  rated  at  per  Ib.  Ans.  5J  cts. 

6.  B  delivered  3  hhds.  of  brandy,  at  6s.   8d.  per  gallon, 
to  C,  for  126  yds.  of  cloth,  what  was  the  cloth  per  yard  ? 

Ans.  10s. 

7.  D  gives  E  250  yards  of  drugget,  at  30  cts.  per  yd. 
for  319  Ibs.  of  pepper ;  what  does  the  pepper  stand  him  in 

er  Ib.  I  Ans.  23  cts.  5ry?. 

8.  A  and  B  bartered  :  A  had  41    cwt.  of  rice,  at   21s* 
yer  cwt.  for  which    B   gave  him   20/.   in   money,  and  tire 
cest  in  sugar  at  8d.  per  Ib.  ;  I  demand  how  much  sugar  B 
gave  A  besides  the  20/.  ?  Ans.  6  cwt.  0  qrs.  I9^lb. 

9.  Two  farmers  bartered :  A'jhad  120  bushels  of  wheat 
at  1£  dols.   per  bushel,  for  which  B  gave  him  100  bushels 
of  barley,  worth  65  cts.  per  bushel,  and  the  balance  in  ^ats 
at  40  cts.  per  bushel ;    what  quantity  of  oats  did  A  receive 
from  B 1  Ans.  287  J  lusheh. 

10.  A  hath  linen  cloth  worth  20d.  an  ell  ready  money  ; 
but  in  barter  he  will  have  2s.  B  hath  broadcloth  worth  14s. 
6d.  per  yard  ready  money  ;  at  what  price  ought   B  to  rate 
bis  broadcloth  in  barter,  so  as  to  be  equivalent  to  A's  bar- 
tering prrcB  ?  An*.  17 s.  4d. 


11.  A  and  B  barter  :  A  hath   145  gallons  qf  Imuid/  ai. 
1  dol.  20  cts.  per  gallon  ready  money,  but  in  barter  he 
will  have  1  dol.  35  cts.  per  gallon  :  B  has  linen  at  58  cts. 
p«r  yard  ready  money  ;  how  must    B    sell  his    linen  per 
yard  in  proportion  to  A's  bartering  price,  and   how   many 
yards  are  equal  to  A's  brandy  1 

Ans.  Barter  price  of  B's  linen  is  65  cts.  2jtfz.  and  he 
must  give  A  300  yds.  for  his  brandy. 

12.  A  has  225yds.  of  shalloon,  at  2s.  ready  money  per 
yard,  which  he  barters  with  B  at  2s.  5d.  per  yard,  taking 
indigo  at  12s.  6d.    per  Ib.   which   is  worth  but   10s.  how 
much  indigo  will  pay  for  the  shalloon ;  and  who   gets  fhs 
best  bargain  ? 

Ans.  43i&.  at  barter  price  will  pay  for  the  shalloon,  and 
B  has  the  advantage  in  barter. 

Value  of  A's  cloth,  at  cash  price,  is  £22     10 

Value  of  43J/6.  of  indigo,  at  10s.  per  Ib.  21     15 

B  gets  the  best  bargain  by     £0     1*5 

LOSS  AND  GAIN, 

IS  a  rule  by  which  merchants  and  traders  discover  their 
profit  or  loss  in  buying  and  selling  their  goods  :  it  also  111*- 
structs  them  how  to  rise  or  fall  in  the  price  of  their  goods, 
so  as  to  gain  or  lose  so  much  per  cent,  or  otherwise. 

Questions  in  this  rule  are  answered  by  the  Rule  of  Three. 
EXAMPLES. 

1.  Bought  a  piece  of  cloth  containing  85  yards,  for  191 
dels.  25  cts.  and  sold  the  same  at  2  dols.  81  cts.  per  yard  ; 
yyhat  is  the  profit  upon  the  -whole  piece  ? 

Ans.  $47,  60  cts. 

2.  Bought  12}  cwt.  of  rice,  at  3  dols.  45  cts.  a  cvvt.    and 
sold  it  again  at  4  €ts.  a  pound ;    what  was  the  whole  gam  ? 

Ans.  $12,  87  cts.  5m. 

3.  Bought  11  cwt.  of  sugar,  at6£d.  per  Ib.  but  could  riot 
sell  it  again  for  any  more  than  2Z.  105.  per  cwt. ;    did  Lgdiii 
Or  lose  by  my  bargain  ?  Ans.  Lost,  £2  11s.  4t£ 

4.  Bought  44  Ib.  of  tea  for  6Z.  12s.  and  sold  it  again  for 
&L  10s.  8vJ.,:  wlrat  I,V>TS  the  crtffil  on  ea'ch  pound  1 

1  An*.  itUrf. 


5.  B'oirgla  a  hhd.  of  molasses  containing  119  gallons,, 
«at  52  cents  per  gallon ;  paid  for  carting  the  same  1  dollar 
23  cents,  and  by  accident  9  gallons  leaked  out ;  at  what 
rate  mfrst  I  sell  the  remainder  per  gallon,  to  gain  13 
in  the  whole  ?  Ans.  69  cts. 


IL  To  know  what  is  gained  or  lost  per  cent. 
RULE. — First  see  what  the  gain  or  loss  is  by  subtraction ;  then,  Aa 
tB"p  price  it  coat :  is  to  the  gain  or  loss^:  :  so  is  100/.  or  $100,  to  this 
gain  or  loss  per  cent. 

EXAMPLES. 

I.  If  I  buy  Irish  linen  at  2s.  per  yard,  and  sell  it  again 
at  #s.  8d.  per  yard  ;  what  do  I  gain  per  cent,  or  in  layin-g 
t}UJt  100/.  :  As  :  2s.  8d.  :  :  IOO/.  :  £33  65.  Qd.  Ans. 

2-.  If  I  buy  broadcloth  at  3  dols.  44  cts.  per  yard,  and  sell 
ft  again  at  4  dols.  30  cts.  per  yard  :  what  do  I  gain  per  ct. 
or  in  laying  out  100  dollars  1 

$  cts.^} 

Sold  for  4, 30  $  cts.     cts.        -$         $ 

Cost       3,  44   >       As  3  44  :  86  :  :  100  ;  25 

Am.  25  per  cent. 
{Jainerd  per  yd.  86  j 

3.  If  I  buy  a  cvvt.  of  cotton  for  34  dols.  86  cts.  and  sell  it 
again  at  41  £  cts.  per  Ib.  what  do  I  gain  or  lose,  and  what 
j>er  cent.  ?  $  cts. 

1  cwt.  at  41^  cts.  per  Ib.  comes  to     46,48 
Prime  co,st    34,86 

Gained  in  the  gross,  $11,61 
As  34,86  :  11,62  :  :  100  :  33J  An*.  33J  per  ccjit. 

4.  Bought  sugar  at  8{d.  per  Ib.  and  sold  it  again  at  4?. 
17s.  per  cwt.  what  did  I  gain  per  cent,  t 

Ans.  £25  19s.  5?</. 

5.  If  I  buy  12  hhds.  of  wine  for  204/.  and  sell  the  saore 
£fgain  at  14J.  17s.  6d.  per  hhd.  do  I  gain  or  lose,  and  w 
per  cent. .?  Ans.  I  lose  12^ per  ccnh 

<>,  At  Hd.  prtffft  in  a  sliflfing,  how  much  per'cenk  ? 

Ate.  £12  JO.?. 


7-  At  25  ets.  profit  in  a  dollar,  how  muesli  p'er  cent  t 


NOTE.  —  When  goods  are  bought  or  sold  on  credit,  you 
must  calculate  (by  discount)  the  present  worth  of  fhfifr 
prire,  in  order  to  find  your  true  gain  or  loss,  &c. 

EXAMPLES. 

1.  Bought  164  yards  of  broadcloth,  at  14s.  Gd.  per  yard 
ready  money,  and  sold  the  same  again  for  154/.  IQs.  on  G 
months  credit  ;  what  did  I  gain  by  the  whole  ;  allowing 
discount  at  G  per  cent,  a  year  ] 

•£.         £.  £.     s.         £.     s. 

As  103  :    100  :  :   154  10    :  150     0  present  worth. 

118  18  prime  cqsK 

Gained  £31     2  Answer. 

&  If  I  buy  cloth  at  4  dols.  1G  cts.  per  yard,  on  eigljt 
months  credit,  and  sell  it  again  at  3  dols.  90  cts.  per  y.d. 
ready  money,  what  do  I  lose  per  cent,  allowing  6  per  cent. 
iTiscount  on  the  purchase  price  '(  -4ns.  2^  per  ccnff 


III.  To  know  how  a  commodity  must  be  sjold,  to  gain 
Or  lose  so  much  per  cent. 

RULE. — As  100  :  is  to  the  purchase  price  :  :  so  is  100J.  or  100 
dollars,  with  the  profit,  added,  or  loss  subtracted  :  to  Ove  selling 
prite. 

EXAMPLES. 

1.  If  I  buy  Irish  linen  at  2s.  3d.  per  yard  ;  how  must  I 
stjl  it  per  yard  to  gain  25  per  cent.  ? 

As  100Z.  :  2s.  3d.  :  :  125/.  to  2s.  9rf.  3  qrs.  Any. 

2.  If  I  buy  rum  at  1  dol.  5  cts.  per  gallon  ;  how  must  I 
&£}  it  per  gallon  to  gain  30  per  cent.  ? 

As  §100  :  $1,05  :  :  $130  :  $l,36£c&.  A~s» 
8.  If  tea  cost  54  cts.  per  Ib. ;  how  must  it  be  sold  per  Ibt 
to  lose  12^  per  cent.  1 

As  $100  :  54  cts.  :  :  $87,  50  cts.  :  47  cts.  2£  m.  An$. 
4.  Bought  cloth  at  17s.  Gd.  per  yard,  which  not  proving 
so  good  as  I  expected,  I  am  obliged  to  lose  15  percent.  bV 
it ;  how  mirst  I  s^ll  it  £er  yard  ?  AT*.  14*. 


5.  Il'l'l  cjy*.  1  qr.  25  Ib.  of  sugar  co'&f  Ii6  dok  50  c;£ 
how  must  it  he  sold  per  Ib.  to  £aiu  30  per  cent,  t 

Ans.  12  cte.  &#. 

G.  Bought  90  gallons  of  wine  at  1  dol.  20  cts.  per  gnll. 
hut  hy  accident  10  gallons  leaked  out ;  at  what  rate  mast  I 
sell  the  remainder  per  gallon  to  gain  upon  the  whole  prime 
co,st>  at  the  rate  of  12  J  per  cent.  I  Ans.  §1,  51  cts.  S^m^ 


IV.  When  there  is  gained  or  lost  pel*   cent,   to 
wfmt  the  commodity  cost. 

RULE. — As  100J.  or  100  dole,  with  the  gain  per  cent,  added,  «r  los? 
per  cent,  subtracted,  is  lo  the  price,  so  is  100  to  the  prime  cost. 

EXAMPLES. 

li  'IF  a  yard  of  cloth  be  sold  at  14s.  7d.  and  there  is  gaiir- 
eti  1W .  13,s.  4d.  per  cent.  ;  what  did  the  yard  cost  I 

£.     s.     d.     s.     d.     £. 
As  116  13     4  :  14    7:  :  100  to   12*.   6d.    Aos. 

2.  By  selling  broadcloth  at  3  dols.  25  cts.    per   yard,  I 
Ipse  at  the  rate  of  20  per  cent.  ;  what  is  the  prime  cost   of 
sakl  cloth  per  yard  ?  Ans.  $4,  06  cts.  2jz?/. 

3.  If  40  Ib.  of  chocolate  be  sold  at  25  cts.  per  Ib.  and  I 
gain  9  per  cent. ;  what  did  the  whole  cost  me  ? 

Ans.  $9,   17  cts.  4w.-f 

4.  Bought  5  cwt.  of  sugar,  and  sold  it  again  at  12  cents 
pfcr  Ib.  by  which  I  gained  at  the   rate    of  25£  per  ceufr.  J 
wllat  did  the  sugar  cost  me  pc#  cwt.  1 

Ans.  $10,  TOcte.  9m<+ 

V.  If  by  wares  sold  at  a  given  rate,  there  is  so  much 
grained  or  lost  per  cent,  to  know  what  would  be  gained  ol- 
lost  per  cent,  if  sold  at  another  rate. 

RULE.— As  the  first  price  :  is  to  100/.  or  100  dols.  with  the  profit- 
per  cent,  added,  or  loss  per  cent,  subtracted  :  :  so  is  the  other  price  •  to 
the  gain  or  loss  per  cent,  at  the  other  rate. 

$.  B.  If  your  answer  exceed  1007.  or  100  dols.  tlra 
excess  is  your  gain  per  cent. ;  but  if  it  be  less  tlran  IOD, 
that  (Jkfici;en<5y  is  tire  loss  per  cent. 


13J. 

x  • .»  . 

EXAMPLES'. 

1.  If  I  seJl  cloth  at  is.  per  yd.  and   thereby  1*4111  15  per 
t.  what  shall  I  gain  per  cent,  if  I  ssll  it  at  (Js,  ppr  yd,  S 

$.       £         s.       £. 
As  5  :  115  :  :  6  :   138         Ans.  gamed  38ptr  ccjit, 

2.  If  I  retail  rum  at  1  dollar  50   cents  per  gallon,  a*n<l 
thereby  gain  25  per  cent,  what  shall  I  gain  or  lose  per  cenU 
if  I  sell  it  at  1  dol.  8  cts.  per  gallon  ? 

$  cts.    $       $  cts.       $ 

1,50 : 125 : :  1,08  :  90       A?is.  I  shall  lose  Wpcr  cent. 

3.  If  I  sell  a  cwt.  of  sugar  for  8  dollars,  and  thereby 
Toss  12  per  cent,  what  shall  I  gain  or  lose  percent,  if  I  sell 
4  cut.  of  the  same  sugar  for  30  dollars  ? 

Ans.  I  lose  evil/  1  per  cent.. 

4.  I  sold  a  watch  for  177.  Is.   5d.  and   by  so  doing  Io4 
}-5'  per  cent,  whereas   I  ought  in   trading  to  have  clenrea 
20  per  cent. ;  how  mnch  wns  it  sold  under  its  rafi-1  v-ahie  ? 

£.      £  s.  d.         £.      £.  s.  d. 
As   85  :  17  1  5  ::  100  :  20  1  8  the  prime  cost. 
mo  :  2-0  1  8  :  :  120  :  24  2  0  the  renl  vaTuei 
Sold  for  17  1  5 

£707    Answer. 


FELLOWSHIP, 

iS  a  rule  by  which  the  accounts  of  several  mel'clrahtS 
or  other  persons  trading  in  partnership,  are  so  ailjiislec^ 
that  each  may  have  his  share  of  the  gain,  or  sustain  his 
share  of  the  loss,  in  proportion  to  his  share  of  the  j-oint 
atock. — Also,  by  this  Rule  a  bankrupt's  estate  nray  be  di- 
vMexi  among  his  creditors,  &c. 

SINGLE  FELLOWSHIP, 

Is  when  the  several  shares  of  stock  arc  coft tinned  in 
trade  an  equal  term  of  time. 

&ULE.— ,AR  tjie  whole  st(yck  is  to  the  \yholc  gnirt  at  losj  •:  so'te  each 
nmn's  par.titfu!aif  stfcsk,  to  his  particular' share  of  Ihe  g -*. 


HE  L  LoVs  HJUP  1*33 

aoF. — Add  ail  Oifc  particular  shares  of  tli6  gain  o*  Icrss  for 
gather,  and  if"  it  b£  right,  the  sum  will  be  equal  t6  the  wfroi'o 
g;aln  or  loss. 

EXAMPLES. 

1.  Two  partners,  A  and  B,  join  their  stock  and  bay 
a  quantity  of  merchandise,-  to  the-  amount  of  820  dollars; 
in  the  purchase  of  which  A  laid  out  350  dollars,  and  B  470 
dollars ;  the  commodity  being  sold,  they  find  their  clear 
gain  amounts  to  250  dollars.  What  is  each  person's  sharp 
of  the  gain  ? 

A  put  in  350 

B  470 


A     OIA     O-A  f  350  :  106,7073+A's  share. 

As  820  :  2*0  :  :     {  47Q  .  143;2926+irs  share. 

Proof  249,9999-4- =$250 

2.  Three  merchants  make  a  joint  stock    of  12007,  cf 
which  A  put  in  240?;  B  3607.  and  C.  600/.  ;  and  by  trading 
they  gain  3257.  what  is  each  one's  part  of  the  gain  1 

Ans.  As  part  £65,  B's  £97  10s.    C's  £162  JOs. 

3.  Three  partners,  A,  B,  and  C,  shipped  108  mules  fnr 
the  West-Indies  ;  of  which  A  owned  48,  B  36,  and  C  24  ; 
But    in  stress  of  weather,   the   mariners  were  obliged  to 
throw  45  of  them   overboard  ;    I  demand  how  much  of 
the  loss  each  owner  must  sustain  1 

Ans.  A  20,1?  15,  and  C  10. 

4.  Four  men  traded  with  a  stock  of  800  dollars,  by 
which  they  gained  307  dols.      A's    stock  was  140  do!s% 
B's  2CO  dols.  C's  300  dols.     I  demand    D's  stock,  and 
what  each  man  gained  by  trading  ? 

Ans.  D's  stock  was  §100,  and  A  gained  $53,  72  cts.  5  m. 
B  §99.  77i  cts.    C  $1 15,  12i  cts.  and  D  $38,  37-J  cts. 

5.  A  bankrupt  is  indebted"  to  A  2117.  to  B  3GO/.  and  to 
C  3917.   and  his  whole  estate  amounts  only  to  6757.  10s. 
which  he  gives  up  to  those  creditors  ;  how  much  must  each 
have  in  proportion  to  his  debt  ? 

Ans.  A  must  Aauc  JG158  05.  3J&  J8£224  135.  4£rf.  and 
C  £2D 


6.  A  Captain,  mate,  and  20  seamen,  took  a  prize 
3501  dols.  of  which  the  captain  takes  11  shares,  and  the 
mate  5  shares;  the  remainder  of  the  prize  is  equally  du- 
vkled  among  the  sailors  ;  how  much  did  each  man  receive? 

3     cts. 

Ans.  The  captain  received       1069,  75 
The  mate  486,  25 

Each  sailor  97,  25 

7*  Divide  the  number  of  360  into  3  parts,  which  shall  be 
fo  each  other  as  2,  3  and  4.  Ans.  80,  120  and  160. 

8.  Two  merchants  have  gained  450/.  of  which  A  is  to 

have  three  times  as  much  as  B  ;  how  much  is  each  to  have? 

Ans.  A  £337  10s.  and  B  £112  10s.— H- 3=4  :  450  :  - 

3  :  £337  lO.s.  A's  share. 

D.  Three  persons  are  to  share  600?.  A  is  to  have  a  cer- 
tain sum,  B  as  much  again  as  A,  and  C  three  times  as* 
much  as  B.  I  demand  each  man's  part  1 

Ans.  A  £66g,  B  j£  133-2,  and  C  £400. 

10.  A  and  B  traded  together  and  gained  100  dols.  A  put 
in  640  dols.  B  put  in  so  much  that  he  must  receive  60  dolsv 
of  the  gain  ;  I  demand  B's  stock  ?  Ans.  $960. 

11.  A,  B  and  C  traded  in  company  :  A  put  in  140  dols, 
B  250  dols.  and  C  put  in  120  yds.  of  cloth,  at  cash  price  ; 
they  gained  230  dols.   of  which  C   took  100  dols.  for  hrs 
share  of  the  gain :  how  did  C  value  his  cloth  per  yard  in. 
common  stock,  and  what  was  A  and  B's  part  of  the  gain! 

Ans.  C  put  in  the,  cloth  at  $%±  per  yard.     A  gamed  $46^ 
67  cts.  6  m.  -\-and  B  £83,  33  cts.  3  w 

MB mg^i  •raa-'JB-'Jiij'  <smtttu.  t- .  r :••**  ^e^^arawMjatwapappaqacqaaat 

COMPOUND  FELLOWSHIP, 

OR  Fellowship  with  time,  is  occasioned  by  several  shared 
of  partners  heing  continued  in  trade  an  unequal  term  of 
time. 

RULE.— Multiply  each  man's  stock,  or  share,  by  the  time  it  was 
continued  in  trade  :  then, 

As  the  sum  of  the  wveral  products, 

Is  to  the  whole  gain  or  loss  : 

&o  is  each  man's  particular  product, 

To  his  purticular  share  of  the  gain  OT  1'cfss? 


•M  POUND  FELLOWSHIP.  1-^5 

EXAMPLES. 

1.  A,  B  and  C  hold  a  pasture  in  common,  for  which  they 
•pay  19/.  per  annum.  A  put  in  8  oxen  for  6  weeks  ;  B  12 
oxen  for  8  weeks  ;  and  C  12  oxen  for  12  weeks  ;  what  must 
each  pay  of  the  rent  2 

Sx  G=  48 
12  x  8=  96 
12  X  12=  144  }>  As  288  : 19*  : : 

Sum  288  3  I  Proof  19    0     0 

.  Two  merchants  traded  in  company;  A  put  in  215 
dols.  for  6  months,  and  B  390  dols.  for  9  months,  but  by 
misfortune  they  lose  200  dols. ;  how  must  they  share  tht? 
loss  ?  Ans.  A's  lass  #53,  75  cts.  JB's  1146,  25  cts. 

3.  Three  persons  had  received  665  dols.  interest:  A  had 
pat  in  4000  dollars  for  12  months,  B  3000  dollars  for  15 
months,  and  C  5000  dollars  for  8  months ;    how  much  is 
each  man's  part  of  the  interest? 

Ans.  A  #240,  B  #225,  and  C  #200. 

4.  Two  partners  gained  by  trading  110/.  12s.  :  A's  stock 
120/.  105.  for  4  months,  and  B's  200Z.  for  61  months  ; 

\vhat  is  each  man's  part  of  the  gain  1 

Ans.  A's  part  £29  18s.3id.}^ff-  B's  £80  iSs.Sld.-fffe 

5.  Two  merchants  enter  into  partnership  for  18  months, 
A  at  first  put  into  stock  500  dollars,  and  at  the  end  of  8 
months  he  put  in  100  dollars  more  ;  B  at  first  put  in  800 
dollars,  and  at  4  months'  end  took  out  200  dols.     At  the 
expiration  of  the  time  they  find  they  have  gained  700  do.l- 
lajs  ;  what  is  each  man's  share  of  the  gain  1 

A        i  #324, 07  4     A's  share. 
'  \  #375,92  5  -B's     do. 

0.  A  and  B  companied  ;  A  put  in  the  first  of  January, 
1000  dollars ;  but  B  could  not  put  in  any  till  the  first  of 
May  ;  what  did  he  then  put  in  to  have  an  equal  share  wirt\ 
A  at  the  year's  end  I 

Mo.  g  Mo.  $ 

As  12    :    1090   :  J    8  ;  1000 x  12m  1500  Am* 


JJ-OUBLE  RULE  OT  T 


DOUBLE  RULE  OF  THREE, 

THE  Double  Rule  of  Three  teaches  to  resolve  at  onc£ 
swch  questions  as  require  two  or  more  statings  in  simpha 
proportion,  whether  direct  or  inverse. 

In  this  rule  there  are  always  five  terms  given  to  find  a 
sixth  ;  the  first  three  terms  of  which  are  a  supposition,  the 
hrst  two  a  demand. 

RULE.  —  In  stating  the  question,  place  the  terms  of  the  supposi- 
tion so  that  the  principal  cause  of  loss,  gain,  or  action,  possess 
the  first  place  ;  that  which  signifies  time,  distance  of  place,  &jck 
in  the  second  place  ;  and  the  remaining  term  in  the  third  place. 
Place  the  terms  of  demand,  under  those  of  the  same  kind  iu 
the  supposition.  If  the  blank  place,  or  term  sought,  fall  un- 
der the  third  term,  the  proportion  is  direct  ;  then  multiply  the 
i'frst  and  second  terms  together  for  a  divisor,  and  the  other  three 
for  a  dividend  :  hut  if  the  blank  fall  under  the  first  or  second 
term,  the  proportion  is  inverse  ;  then  multiply  the  third  and 
fourth  terms  together  for  a  divisor,  and  the  other  three  for  a  di» 
vrdcnd,  and  the  quotient  will  be  the  answer. 

EXAMPLES. 

1.  If  7  men  can  build  36  rods  of  wall  in  3  days  ;  how 
many  rods  can  20  men  build  in  14  days  1 

7  :     3  :  :  36  Terms  of  supposition 
20  :  14  Terms  of  demand> 

36 


0 

* 


42  .//c 

504 
20 

7  X  3=21)10080(480  rods.  Ans, 

2.  If  100Z.  principal  will  gain  6Z.  interest  in 
what  will  400/.  gain  in  7  months  ? 

Principal  1007.  :  12  ma.  :  :  6/-.  interest. 
400     :     7  Jtos, 


3.  If  100?.  will  gain  61  a  year ;  in  what  time  will  4W. 
gain  14/.  £•       mo.         £ 

100    :  1-2  :  :    6 

400    :         :  :  14     Ans.  7  months. 

4.  If  400/.  gain  14/.  111  7  months  :  what  is   the  rate  pef 
Cent,  per  annum  1  £.       mo.       Int. 

400    :     7  :  :  14 
100    :   12  Ans.  £6. 

3.  What  principal  at  6/.  per  cent,  per  annum,  will  give 
147.  in  7  months  1  £.       mo.        Int. 

100    :   1*2  :  :     6 

7  :  :  14  Ans.  £400. 

6.  An  usurer  put  out  86?.  to  receive  interest  for  the  same  ; 
and  when  it  had  continued  8  months,  he  received  principal 
and  interest,  SSL  17s.  4d.  ;  I  demand  at  what  rate  per  ct. 
per  ami.  he  received  interest?  Ans.  5 per  cent. 

7.  If  20  bushels  of  wheat  are  sufficient  for  a  family  at* 
8  persons  5  months,  how  much  will  be  sufficient  for  4  per* 
'sons  12  months  7  Ans.  24  bushels. 

8.  If  30  men  perform  a  piece  of  work  in  20  days  ;    how 
many  men  will  accomplish  another  piece  of  work   4  time's 
;vs  large  in  a  fifth  part  of  the  time  ? 

80  :  20  :  :   1 

4  :  :  4  Ans.  COO. 

9.  If  the  carriage  of  5  cwt.   3   qrs.    150    miles,  cost   24 
dollars  58  cents ;  what  must  be  paid  for  the  carriage   of  7 
xjwt.  2  qrs.  25  Ib.  64  miles,  at  the  same  rate  1 

Ans.$U,QScts.  6m.  + 

10.  If  8  men  can  build  a  wall  20  feet  long,  6  feet  high* 
and  4  feet  thick,  in  12  days  ;    in  what   time  will  24  men 
build  one  200  feet  long,  8  feet  high,  and  6  feet  thick  1 

8  :   12  :  :  20x6x4 


24  :  200  x  8  x  6  80  days.  Ans. 


CONJOINED    PROPORTION, 

IS  when  the  coins,  weights  or  measures  of  several  coun- 
tries are  compared  in  the  same  question  ;  or  it  is  joining 
ntany  proportions  together,  and  by  the  relation  wh&h 


C  OX  JOINED    PROPOft  T  JO  X'. 

several  antecedents  have  to  their  consequents,  the  prop^iv 
i'ion  between  the  first  antecedent  and  the  last  consequent  is 
discovered,  as  well  as  the  proportion  between  the  others  in 
their  several  respects. 

NOTE. — This  rule  may  generally  he  abridged  by  can- 
celling equal  quantities,  or  terms  that  happen  to  be  the 
same  in  both  columns  :  and  it  may  be  proved  by  as  many 
statings  in  the  Single  Rule  of*  Three  as  the  nature  of  the 
question  may  require. 

CASE  I. 

When  it  is  required  to  find  how  many  of  the  first  st)rt 
of  coin,  weight  or  measure,  mentioned  in  the  question,  are 
equal  tq  a  given  quantity  of  the  last. 

RULE. — Place  the  numbers  alternately,  beginning  at  the  left  hand, 
and  let  the  last  number  stand  on  the  left  hand  column  ;  then  multi- 
ply the  left  hand  column  continually  for  a  dividend,  and  the  right 
hand  for  a  divisor,  and  the  quotient  will  be  the  answer. 

EXAMPLES. 

1.  If  100  lb.  English  make  95  Ib.  Flemish,  and    19  Ih. 
Flemish  25  lb.  at  Bologna ;  how  many  pounds  English  are 
equal  to  50  lb.  at  Bologna? 

*  Ib.  lb. 

100  Eng.=95  Flemish. 
19  Fie.  =25  Bologna. 
50  Bologna.  Then  95  X  25==2375  the  divistm 

S5000  dividend,  and  2375)95000(40  Ans. 

2.  If  40  lb.  at  New-York  make  48  lb.  at  Antwerp,  and 
30  lb.  at  Antwerp  make  36  lb.  at  Leghorn  ;    how  many  Ih. 
at  New- York  are  equal  to  144  lb.  at  Leghorn  ? 

Ans.  100/6. 

3.  If  70  braces  at  Venice  be  equal  to  75  braces  at.Lteg- 
Iiorn,  and  7  braces   at  Leghorn  be  equal  to  4  American 
yards ;  how  many  braces  at  Venice  are  equal  to  64  Ameri- 
"cao  yards?  Ans.  104^. 

CASE  II. 

When  it  is  required  to  find  how  many  of  the  last  sort  of 
ct)in,  weight  or  measure,  mentioned  in  the  question-,  are 
equal  to  a  given  quantity  of  the 


uLv— Place  the  numbers  alternately,  beginning  at  the  left  hantfc, 
Und  let  the  last  number  stand  on  the  right  hand  ;  then  multiply  Cre 
first  row  for  a  divisor,  and  the  second  for  a  dividend. 

EXAMPLES. 

1.  If  24  Ib.  at  New-London  make  20  Ib.  at  Amsterdam-, 
and  50  Ib.  at.  Amsterdam  60  Ib.  at  Paris ;    how  many  at 
Paris  are  equal  to  40  at  New-London  ? 
Left.      Right. 
24  =  20        20   x   60   x  »40  =  48000 

50   =  60  ±=    40   Ans. 

40        24  x  50  =  1200 

&.  If  50  Ib.  at  New-York  make  45  at  Amsterdam,  and 
80  Ib.  at  Amsterdam  make  103  at  Dantzic  ;  how  many  Ib* 
at  Dantzic  are  equal  to  240  at  N.  York  ?  Ans.  278TV 

3.  If  20  braces  at  Leghorn  be  equal  to  1 1  vares  at  Lis- 
bon, and  40  vares  at  Lisbon  to  80  braces  at  Lucca ;  how 
jriany  braces  at  Lucca  are  equal  to  100  braces  at  Leghorn '{ 

Ans.  110. 


EXCHANGE. 

BY  this  rule  merchants  know  what  sum  of  money  onght 
to  be  received  in  one  country,  for  any  sum  of  different  spe- 
cie paid  in  another,  according  to  the  given  course  of  ex- 
change. 

To  reduce  the  moneys  of  foreign  nations  to  that  of  tire 
United  States,  you  may  consult  the  following 

TABLE : 
Showing  the  value  of  the  moneys  of  account,  of  foreign 

nations,  estimated  in  Federal  money.*     $  cts. 
Pound  Sterling  of  Great  Britain,  4  44 

Pound  Sterling  of  Ireland,  4  10 

lavre  of  France,  0  18 1 

Guilder  or  Florin  of  the  U.  Netherlands,  0  39  * 

Mark  Banco  of  Hamburgh,  0  3<U 

Rix  Dollar  of  Denmark,  1     Q3 

*  taws  tT.  S,  A. 


HO  , 

liial  Plate  of  Spain,  0'  10 

Mil  re  a  of  Portugal,  1  24 

Tale  of  China,  1  48 

Pagoda  of  India,  1   94 

Rupee  of  Bengal,  0  55J. 

L— OF  GREAT  BRITAIN. 

EXAMPLES. 

1.  In  45?.  10s.  sterling,  how  many  dollars  and  cents  ? 

A  pound  sterling  being=444  cents, 
Therefore— As  II  :  444  cts.  :  :  45,5/.  :  20202  cts.  Ans. 

2.  In  500  dollars  how  many  pounds  sterling? 

As  444^5.  :1Z.::  50000  cfe.  :  112/.  12s.  3rf.+    Arts. 
II.— OF  IRELAND. 

EXAMPLES. 

1.  In  907.  10s.  6d.  Irish  money,  how  many  cents  ? 

II  Irish=410  c/5. 

£.     cts.         £.  cts.  $     cts. 

Therefore— As  1  :  410  :  :  90,525    :    37115T=371,  15] 

2.  Tn  168  dols.  10  cts.  how  many  pounds  Irish? 
As410cfe.  :   17.  :  :   16810  cts.  :  £41  Irish.         Arts, 

HI.— OF  FRANCE. 

Accounts  are  kept  in  livres,  sols  nnd  deniers. 
4  12  deniers,  or  pence,  make    I  sol,  or  shilling, 
\  20  sols,  or  shillings,         —     1  livre,  or  pound. 

EXAMPLES. 

1.  In  250  livres,  8  sols,  how  many  dollars  and  cents, 
1  livre  of  France  —18^  cts.  or  185  mills. 

£.       m.  £.  m.  $  cts.  m. 

Aa  1  :  185  :  :  250,4  :  46324  =  46  32  4          Ans. 

2.  Reduce  87  dols.  45  cts.  7  m.  into  livres  of  Franco. 
mills,  liv.          mills.       liv.     so.  den. 

As  185  :   1  :  :    87457  :  472    14  9+        Am. 
IV.— OF  THE  TJ.  NETHERLANDS. 
Accounts  are  kept  here  in  guilders,    stivers,  groats  and 

. 

f    8  phennings  make  1  groat. 
<    2  groats  —      1  stiver. 

(  20  stivers  - —      1  guilder  or  florim 

A  guilder  i£— #9  ctfnts,  flr  390  mills. 


i;xcHA:\ci:  111 

EXAMPLES. 

Reduce  124  guilders,  14  stivers,  into  federal  money. 

Guil       cts.  Guil          $    *d.     c.     m. 

As  1     :     39    ::    124,7     :    48,  6      3      3  An* 

mills.     G.         mills.          G. 
As  390  :    1    :  :    48633  :  124,7  Proof. 

V.— OF  HAMBURGH,  IN  GERMANY. 

Accounts  are  kept  in  Hamburgh  in  marks,  sous  and  dp- 
tiers'-liibs,  and  by  some  in  rix  dollars. 

C  12  deniers-lubs  make  1  sous-lubs. 

<  16  sous-lubs,       —      1  mark-luhs. 

\    3  mark-lubs,     —      1  rix  dollar. 
NOTE. — A  mark  is  =  33^  cts.  or  just  \  of  a  dollar. 
HULE.— Divide  the  marks  by  3,  the  quotient  will 

EXAMPLES. 

Reduce  641  marks,  8  sous,  to  federal  money, 
3)641,5 


$213,833  Ans. 

But  to  reduce  federal  money  into  marks,  multiply  flfp 
given  sum  by  3,  &c. 

EXAMPLES. 

Reduce  121  dollars,  90  cts.  into  marks  banco, 
121,90 
3 

565,70=365  marks,  11  sous,  2,4  den.  AT&. 
VL— -OF  SPAIN. 

Accounts  are  kept  in  Spain  in  piastres,  rials,  and  nrafc- 
\adieg. 

!34  marvadies  of  plate  make  1  rial  of  plate. 
8  rials  of  plate  —    1  piastre  or  piece  of  8, 

To  reduce  rials  of  plate  to  federal  money. 
Since  a  rial  of  plate  is  =  10  cents  or  1  dime,  you  need 
QDly  call  the  rials  so  many  dimes,  and  it  is  done» 
EXAMPLES. 


Bnt  to  reduce  cents  into  rials  of  plate,  divide  by  10  ; 
Thus,  845  cents ~-J 0=84,5=84  rials,  17  marvadies,  &c. 

VII.— OF  PORTUGAL. 

Accounts  are  kept  throughout  this  kingdom  in  milreal^, 
a*nd  reas,  reckoning  1000  reas  to  a  milrea. 

NOTE. — A  milrea  is  =  124  cents  ;  therefore  to  reduce 
milreas  into  federal  money,  multiply  by  124,  and  the  pro- 
duct will  be  cents,  and  decimals  of  a  cent, 

EXAMPLES. 

1.  In  340  mil  reas  how  many  cents? 

340  x  124=42160  cents=$421,  60  cts.  An* 

&  In  211  mil  reas,  48  reas,  how  many  cents'? 

NOTE. — When  the  reas  are  less  thnn  100,  place  a  cipher 
before  them.— Thus,  211,048  x  124=26169,952  cts.  or  261 
dols.  69  cts.  9  mills.  4-  Ans. 

But  to  reduce  cents  into  milreas,  divide  them  by  124$ 
and  if  decimals  arise  you  must  carry  on  the  quotient  as  far 
as  three  decimal  places  ;  then  the  whole  numbers  thereof 
Will  be  the  milreas,  and  the  decimals  will  be  the  reajs, 

EXAMPLES. 
X.  In  4195  cents,  how  many  milreas? 

4195—124=33,830  ~  or  33  milreas,  830  reas.  Ans. 
£.  In  24  dols.  92  cents,  how  many  milreas  of  Portual] 

Ans.  20  milreas,  096  rea& 
VIII.— EAST-INDIA  MONEY. 
To  reduce  India  Money  to  Federal,  viz. 
f  Tales  of  China,  multiply  with  148 

<  Pagodas  of  India,  '194 

(  Rupee  of  Bengal,  55^ 

EXAMPLES. 

1.  In  641  Tales  of  China,  how  many  c^nts? 

Ans.  94868 

2.  In  50  Pagodas  of  India,  how  many  cents  ? 

Ans.  9708 

3.  In  98  Rupeies  of  Bengal,  how  many  cents  I 

An*. 


VULGAR  FRACTION. 

HAVING  briefly  introduced  Vulgar  Fractions  innnj> 
tfrately  after  reduction  of  whole  numbers,  and  given  somd 
general  definitions,  and  a,  few  such  problems  therein  as 
\yere  necessary  to  prepare  and  lead  the  scholar  immediate- 
ly to  decimals ;  the  learner  is  therefore  requested  to  nrotl 
those  general  definitions  in  page  69. 

Vulgar  Fractions  are  either  proper,  improper,  single* 
Compound,  or  mixed. 

1.  A  single,  simple,  or  proper  fraction,  is  when  the  nu> 
tnerator  is  less  than  the  denominator,  as  •£,  J,  f ,  -£•£-,  &•€.- 

2.  An  Improper  Fraction,   \s   when    the   numerator  ex-*- 
tieexls  tlie  denominator,  as  3,  J,  ^ ,  &c. 

3.  A  Compound  Fraction,  is  the  fraction   of  a   fraction, 
Coupled  by  the  word  of,  thus,  -*-  of  Tljt  |  of  §  of  J,  &c. 

4.  A  fifixed  Number,  is  composed  of  a  whole  number  nntf 
•a  fraction,  thus,  8|,  14 ^,  &c. 

5.  Aii3r  whole  number  may  be  expressed  like  a  fractrorc 
by  dravving  a  line  under  it,  and  putting  1  for  denominator,, 
thus,  8==f,  and  12  thus,  y,  &c. 

6.  The  common  measure  of  two  or  more  numbers,  is 
that  number  which  will  divide  each  of  them  without  a  re- 
mainder ;  thus,  3  is  the  common  measure  of  12,  24,  and  3Q.J 
and  the  greatest  number  which  will  do  this   is  called   tire 
greatest  common  measure. 

7.  A' number,  which  can  be  measured  by  two  or  more 
numbers,  is  called  their  common  multiple :  and  if  it  be  the 
least  number  that  can  be  so  measured,  it  is  called  the  least 
common  multiple :  thus  24  is  the  common  multiple  2,  3  and 
4 ;  but  their  least  common  multiple  is  12. 

To  fmdjihe  least  common  multiple  of  two  or  more  num- 
bers. 

RUT  E. — 1.  Divide  by  any  number  that  will  divide  two  or  more  of 
the  given  numbers  without  a  remainder,  and  set  the  quotients,  tog.e^ 
ther  with  the  undivided  numbers,  in  a  line  beneath. 

2.  Divide  the  second  liner-'  as  bsforo,  and  so  on  till  there  are  no  two- 
numbers  that  can  be  divided  ;  then  the  continued  product  of  tire  di-. 
vistfrs  and  quotients,  will  give  the  multiple  remii: 


144  Bj^uroiro'ft  OP  vo- 


1.  What  is  the  least  common  multiple  of  4,  S,  6  and  10 1 
X5)4    5    6     10 

X2)4     1     6      Q 
X2     1x3      1 


5x2x2x3— m  An  A 
S.  What  is  the  common  multiple  of  6  and  SI 

Ans.  24. 

3.  What  is  the  least  number  that  3,  5,  8  and   12  will 
measure  ?  ^7W.  130. 

4.  What  is  the  least  number  that  can  be  divided  by  tht? 
9  digits  separately,  without  a  remainder  2        Ans.  252I>. 


DEDUCTION  OF  VULGAR  FRACTIONS, 

IS  the  bringing  them  out  of  one  form  into  another,  in  o*iv 
cfer  to  prepare  them  for  the  operation  of  Addition,  Sub- 
traction, &c. 

CASE  I. 
To  abbreviate  or  reduce  fractions  to  their  lowest-teTms*. 

RULE. — 1.  Find  a  common  measure,  by  dividing  the  greater  terjrt 
by  the  le«s,  and  this  divisor  by  the  remainder,  and  so  on,  always  di-f 
viding  the  last  divisor  by  the  last  remainder,  till  nothing  remains^; 
the  laft  divisor  is  the  common  measure.* 

2.  Divide  both  of  the  terms  of  the  fraction  by  the  common  nroa- 
sure,  and  the  quotients  will  make  the  fraction  required. 


*  To  find  the  greatest  common  measure  of  more  than  two  numbers,  you, 
must  find  the  greatest  corr.mon  measure  of  two  of  them  as  per  rule  above  j 
then,  of  that  common  measure  and  one  of  the  other  numbers,  and  so  on 
through  all  the  numbers  to  the  last ;  then  will  the  greatest  ccimiroir  mea- 
sure last  tfwiM  he  the 


GN  OF  VULGAR  FRACT.IQN3.  145 

OK,  it*  3^1  chogp^,  yo,u  may  take  th&t  eti£y  m£tho,d  in  Brobfgth  I. 
age  62.) 

EXAMPLES; 

1.  Reduce  ff  to  its  lowest  terms. 

48)f|(i   "  Operation. 

•sT\48/6  common  measure,  8)f$=v$  Ans. 

/tfv    Tlcpi 

OfiT        **y/«-» 

%.  Reduce  $f  to  its  lowest  terms.  Ans.  £& 

3.  Reduce  |£|  to  its  lowest  terms.  Ans.  i| 

4.  Reduce  -f  Jg-f  to  its  lowest  terms.  Ans.  % 

CASE  II. 

To  reduce  a  mixed  number  to  its  equivalent  improper 
fraction. 

RULE.  —  Multiply  the  whole  number  by  the  denominator  o'f  thfe  gi- 
ven fraction,  and  to  the  product  add  the  numerator,  this  surti  written 
above  the  denominator  will  form  the  fraction 


1.  Reduce  45  J  to  its  equivalent  improper  fraction. 


2.  Reduce  19}|  to  its  equivalent  improper  fraction. 

Ans.  \*/ 

3.  Reduce  l^yW  to  an  improper  fraction. 

Ans.  t-V^8 

4.  Reduce  6l£f|  to  its  equivalent  improper  fraction. 

Ans.  «ff|5 
CASE  III. 

To  find  the  value  of  an  improper  fraction. 
RULE.  —  Divide  the  numerator  by  the  denominator,  and  the  qur? 
tient  will  be  the  value  sought. 

EXAMPLES. 

ANSWERS, 

1.  Find  the  value  of  y  5)48(9f  * 

2.  Find  the  value  of  3Ty  !9}| 

3.  Find  the  value  of  9T3T3 

4.  Fifid  the  value  of  2|§S5 

5.  Find  the  value  of  V 

N 


146  REDUCTION  OF  VULGAR 

CASE  IV. 

To  reduce  a  whole  number  to  an  equivalent  fraction^  Irav  - 
ing  a  given  denominator. 

RULE. — Multiply  the  whole  number  by  the  given  denominator  ;• 
place  the  product  over  the  said  denominator,  and  it  will  form  the 
fraction  required. 

EXAMPLES. 

1.  Reduce  7  to  a  fraction  whose  denominator  will  be  9. 

Thus,  7x9=63,  and  V  the  Ans. 

2.  Reduce  18  to  a  fraction  whose  denominator  shall  be 
12.  Ans.  a-f-f 

3.  Reduce  100  to  its  equivalent  fraction,  having  90  fbr  a 
.denominator.  Ans.  9fJ°=9J°=1  J° 

CASE  V. 

To  reduce  a  compound  fraction  to  a  simple  one  of  equal 
value. 

RULE. — 1.  Reduce  all  whole  and  mixed  numbers  to  their  equiva- 
lent fractions. 

2.  Multiply  all  the  numerators  together  for  a  new  numerator,  and 
all  the  denominators  for  a  new  denominator ;  and  they  will  f 
fraction  required. 

EXAMPLES. 

1.  Reduce  J-  off  of  £  of  TVto  a  simple  fraction. 
1x2x3x4 


2x3x4x10 

2.  Reduce  f  of  £  of  f  to  a  single  fraction.      Ans. 

3.  Reduce  f  of  ji  of  ^f  to  a  single  fraction. 

Ans. 

4.  Reduce  f  of  |  of  8  to  a  simple  fraction. 

Ans.        =3 

5.  Reduce  |  of  Jf  of  42£  to  a  simple  fraction. 

Ans.  'fjr 

NOTE. — If  the  denominator  of  any  member  of  a  com- 
pound fracti'on  be  equal  to  the  numerator  of  another  mem- 


REDUCTION  Or  VULGAR  FRACTIONS.        147 

ber  thereof,  they  may  both  be  expunged,  and  the  other 
members  continually  multiplied  (as  by  the  rule)  will  pro- 
duce the  fraction  required  in  lower  terms. 
(3.  Reduce  f  off  off  to  a  simple  fraction. 
Thus  2x5 

—=£*=&  An*. 
4X7 

7.  Reduce  -J  off  off  of  {4  to  a  simple  fraction. 

Ans.^=-ll 

CASE  VI. 

To  reduce  fractions  of  different  denominations  to  equiva- 
lent fractions  having  a  common  denominator. 

RULE  I. 

1.  Reduce  all  fractions  to  simple  terms. 

2.  Multiply  each  numerator  into  all  the  denominators  except  its? 
own,  for  a  new  numerator;  and  all  the  denominators  into  each  other 
continually  for  a  common  denominator ;  this  written  under  the  seve- 
ral ire\v  numerators  will  give  the  fractions  required. 

EXAMPLES. 

1.  Reduce  |-,  f ,  J,  to  equivalent  fractions,  having  a  coni,- 
irron  denominator. 

-J  +  -|  -f  -4— 24  common  denominator. 


1 

2 

3 

% 

x3 

2 

3 

3 

4 

0 

:<4 

4 

2 

12 

!&:•• 

18 

new  numerators. 

24 

24 

24 

denominators. 

2.  Reduce  f,  -,%,  and  |4,  to  a  common  denominator. 


3.  Reduce  |,  £,  f  ,  and  |,  to  'a  common  denominator. 

Ans,     f  f  .  4ff  ,  Iff,  flu* 


4'8  REDUCTION  0  F  V  U  LG  AR  FR  At  T  iUX  b.. 

4.  Reduce  -,  -?»  ,  and  T\,  to  a  common  denominator^ 
800         300"        400 


1000   1000   1000 

&  Reduce  J,  £>  and  12^3  tp  a  common  denominator. 

^•«,  «,  W- 
6.  Reduce  f,  f,  and  £  of  |i,  to  a  common  denominator. 

' 


The  fore^in^  is  a  gemeral  rule  for  reducing  fractions  to 
a  comroou  denominator  ;  but  as  it  will  save  much  labour  to 
keep  the  fractions  in  the  lowest  terms  possible,  the  follow- 
ing Rule  is  much  preferable. 

RULE  II. 

For  reducing  fractions  to  the  least  common  denominator. 
(By  Rule,  page  143)  find  the  least  common  multiple  of 
all  the  denominators  of  the  given  fractions,  and  it  will  be 
the  common  denominator  required,  in  which  divide  each 
particular  denominator,  and  multiply  the  quotient  by  its 
own  numerator,  for  a  new  numerator,  and  the  new  nume- 
rators being  placed  over  the  common  denominator,  will  ex- 
press the  fractions  required  in  their  lowest  terms. 

EXAMPLES. 

1.  Reduce  ^  f  ,and  |  ,  to  their  iftsf  common  denominator, 
4)2    4    8 


111     4x£=r8  the  least  com.  denominator. 


S— 2x1=4  the  1st  numerator. 
.8—4x3=6  the  2d  numerator. 
8— $X5=5  the  3d  numerator, 
numbers  placed  over  the  denominator,  give  (he 
tmswer  f- ,  f ,  f ,  equal   in  value,  and  in  much  lower  terms 
than  the  general  Rule  would  produce  ff,  f  f ,  f  £. 

2i  ll-editee  f ,  f ,  and  TV,  to  their  least  common  denomina- 
tor. An*.  U,  4f ,  4ft. 


REDLTCtLQ.V  OF  VULGAIi  FKA.C.Tl'  149 

tf.  Reduce  i  f  f  and  -^  to  their  least  common  denomi- 
nator. Ans.  -J3-  -/T  if  J-f 

4.  Reduce  I  J  |-  and  ^  to  their  least  common  denomi- 
nator. Ans.  &  If  }|-  W 

CASE  VII. 

To  Reduce  the  fraction  of  one  denomination  to  the  frac- 
tion of  another,  retaining  the  same  value. 


Reduce  the  given  fraction  to  such  a  compound  one,  as 
will  express  the  value  of  the  given  fraction,  by  comparing 
it  with  all  the  denominations  between  it  and  that  denomi- 
nation you  would  reduce  it  to  ;  lastly,  reduce  this  com- 
pound fraction  to  a  single  one,  by  Case  Y. 

EXAMPLES. 

1.  Reduce  |  of  a  penny  to  the  fraction  of  a  pound* 
By  comparing  it,  it  becomes  f  of  VV  of  ^-0-  of  a  poun'd. 
5  x  1  x  1  5 


6x  12x20  1440 

2.  Reduce  TT5T^  of  a  pound  to  the  fraction  of  a  penny. 

Compared  thus  T/T77  of  2T°  of  yd. 
Then         5  x  20  x  12 


1440         1          1 

3.  Reduce  ^  of  a  farthing  to  the  fraction  of  a  shilling. 

Ans.  ij 

4.  Reduce  f  of  a  shilling  to  the  fraction  of  a  pound. 

Ans.  7f  ¥-^yQ 

5.  Reduce  4  of  a  pwt.  to  the  fraction  of  a  pound  troy. 

AllS.   T/Jo-— 3"3F 

6.  Reduce  f  of  a  pound  avoirdupois  to  the  fraction  of  a 
cwt. 

7.  What  part  of  a  pound  avoirdupois  is  T  JF  of  a  cwt. 
Compounded  thus  T|¥  off  of  V=iW =# 

8.  What  part  of  an  hour  is  ^  of  a  week. 


•60  RKDl'i.TIuN    OF   VULGAR   i;HACT  1O 

9.  Reduce  f  of  a  pint  to  the  fraction  of  a  hhd.  Ans.  ^-2- 

10.  Rediice  J  of  a  pound  to  the  fraction  of  a  guinea. 

Compounded  thus,  J  of  2T°  of  ^VS>==:T 

11.  Express  5J-  furlongs  in  the  fraction  of  a  mile. 

Thus  5J=V  of  i=H 

12.  Reduce  f  of  ail  English  crown,  at  6s.  8d.  to  the  fjjac- 
t?0n  of  a  guinea  at  28s.  Ans.  -£T  o/  rt  guinea. 

CASE  Vllt. 

T!o  find  the  value  of  a  fraction  in  the  known  parts  of  the 
integer,  as  of  coin,  weight,  measure,  &e. 

RULE. 

Multiply  the  numerator  hy  the  parts  in  the  next  inferior 
denomination,  and  divide  the  product  by  the  denominator  ; 
$nd  if  any  trthig  remains,  multiply  it  by  the  next  inferior  de* 
nomination,  and  divide  by  the  denominator  as  before,  and  so 
on  as  far  as  necessary,  and  the  quotient  will  be  the  answer. 

NOTE.  —  This  and  the  following  Case  are  the  same  with 
^Problems  II.  and  III.  pages  TO  and  71  ;  but  for  the  scho- 
lar^s  exercise,  I  shall  give  a  few  more  examples  in  each. 

EXAMPLES. 

1.  What  is  the  value  of  f  11  of  a  p6und  ?  Ans.  8s.  9£d. 
£.  Ifind  tlie  value  of  £  of  a  c\vt.  Ans.  3  qrs.  3  Ib.  1  oz.Vty  dr. 
&.  Bind  the  value  off  of  3s.  6d.  Ans.  &s. 


4.  B^w  imrch  is  y6^-  of  a  pound  avoirdupois  ? 

Ans.  7  ov.  10  d'r, 


&.  How  mucfe  is  f  of  a  hhd.  of  wine  ?       Ans. 

6.  What  is  the  value  of  |f  of  a  dollar  ?    Ans.  5*5. 

7).  'What  is  the  value  of  T\  of  a  guinea  ?          Arts.  tSs\ 


ADDITION  OF  VULGAR  F.RACTIONS.  151 

8.  Required  the  value  of  £J-J  of  a  pound  apothecaries. 

Ans.  2  oz.  3  grs. 

9.  How  much  is  |  of  57.  9s.  1  Ans.  £4  13s.  5|rf. 
10.  How  much  is  -»-  of  f  of  J  of  a  hhd.  of  wine  t 

-4^5.  15  gals.  3  ??fc. 

CASE  IX. 

To  reduce  any  given  quantity  to  the  fraction  of  any  greater 

denomination  of  the  same  kind. 
[See  the  Rule  in  Problem  III.  page  71.] 

EXAMPLES  FOR  EXERCISE. 

1.  Reduce  12  Ib.  3  oz.  to  the  fraction  of  a  cwt. 

4K*lWjf 

2.  Reduce  13  cwt.  3  qrs.  20  Ib.  to  the  fraction  of  a  ton. 

Ans.  || 

3.  Reduce  16s.  to  the  fraction  of  a  guinea.        Ans.  -| 

4.  Reduce  1  hhd.  4$  gals,   of  wine  to  the  fraction  of  a 
tun.  t  Ans.  % 

5.  What  part  of  4  cwt.  1  qr.  24  Ib.  is  3  cwt.  3  qrs.  17  Ib. 
8  oz».  ?  Ans.  I 


ADDITION  OF  VULGAR  FRACTIONS. 

RULE. 

Reduce  compound  fractions  to  single  ones  ;  mixed  num- 
bers to  improper  fractions ;    and  all  of  them  to  their  least 
common  denominator,  (by  Case  VI.  Rule  II.)  then  the  sum 
of  the  numerators  written  over  the  common  denominator 
;II1  be  the  sum  f>f  the  fractions  required. 

EXAMPLES. 

1.  Aild  Bj-  J  and  f  off  together. 

5£=y  and  f  of  f— J| 
Then  y ,  J,  4J  reduced  to  their  least  common  denominator 

by  Ca'se  VI.  Rule  If.  will  become  W,  £f*  H 
Then  13^-f-  I8-V  14r=  W  =0,14  or  G%    Ans\ 


152  ADDITION  OF  VULGAR 

2.  Add  |,  £  ,  and  $  together.  ANSWERS.  1J 

3.  Add  i,  J,  and  f  together.  1  1 
4-  Add  l2i  3f  and  4  £  together.  20  J4 
5.  Add  J  of  95  and  $  of  14  J-  together.  44j| 

NOTE  1.  —  In  adding  mixed  numbers  that  are  not  com- 
pounded with  other  fractions,  you  may  first  find  the  sum  of 
the  fractions,  to  which  add  the  whole  numbers  of  the  given 
mixed  numbers. 

6-  Find  the  sum  of  5J,  7f  and  15. 

I  find  the  sain  of  J  and  £  to  be 
Then  lll+5+7 

7.  Add  f  and  17^  together.  ANSWERS.  17-^- 

8.  Add  25,  8}  and  -J-  of  |  of  -/0  33^ 

NOTE  2.  —  To  add  fractions  of  money,  weight,  &c.  reduce 
fractions  of  different  integers  to  those  of  the  same. 

Qr,  if  you  please,  you  may  find  the  value  of  each  fraction 
e  VTII1.  in  Reduction,  and  then  add  tliem  in  their 
yroper  terms. 

9.  Add  -\  of  a  shilling  to  f  of  a  pound, 

1st  method  2d  method. 

4  of  *V=T*T£-  »^=7s.  Cd.  Oqrs. 

Then  TH+f-T4f¥^.  1  ^-=0     6    3f 

Whole  value  by  Case  VIII. 


is  8s.  Od.  3f  qrs.  Arts.  Ans.  8     0     3  J 

By  Case  VIII.  Reduction'. 

10.  Add  |  Ib.  Ti?oy,  to  |  of  a  pwt. 

Ans.  7o 

11.  Add  4  of  a  ton,  to  T°^  of  a  cwt. 

Ans.  12  cz^.  1  gr.  8 

12.  Add  f  of  a  mile  to  T\  of  a  furlong.  Ans. 

13.  Add  ^  of  a  yard,  £  of  a  foot,  and  J-  of  atiib  together. 

^is.  1540  y^s.  g  ^f.  9  in. 

14.  Add  J  of  a  week,  £  of  a  day,  £  of  An  hour,  and  £  of 

fe.  2  ^/«,  2  7w*  3f)  ??F^  45  5ert 


SUBTRACTION  OF  VUL.GAR  FRACTIONS.  163 

SUBTRACTION  OF  VULGAR  FRACTIONS. 
RULE.* 

Ptepare  the  fraction  as  in  Addition,  and  the  difference 
of  the  numerators  written  above  the  eommon  denominator, 
will  give  Ihe  difference  of  the  fraction  required. 

EXAMPLES 

1.  From  $  take  f  of  £ 

|  of  J=M=T7§-  Then  a  and  T\=rV  TV 

Therefore  9 — 7=ff=-J  */<£  -4ns. 

2.  From  |f  take  4  /Insiders.  Ji 

3.  From  }-J  take  T%  ^VV 

4.  From  i  4  take  ]  f  13  r^ 

5.  What  is  the  difference  of  T\  and  |-f  ?  ir'H 

6.  What  differs  TV  from  -J  1  TW 

7.  From  14^  take  |  of  19  1TV 

8.  From  f  i  take i]i  0 remains. 

9.  From  |4  of  a  pound,  take  \  of  a  shilling. 

«  of  ¥i_=rj7r£.     Then  from  }i£.  take  T|o^.  ^W5.  f|£. 

NOTE. — In  fractious  of  money,  weight,  &c.  you  may,  if 
you  please,  find  the  value  of  the  given  fractions  (by  Case 
VIII.  in  Reduction)  and  then  subtract  them  in  their  proper 
terms. 

10.  From  TV&.  take  3J  shillings.    -4ns.  5s.  (yd.  2|  qrs. 

11.  From  f  of  an  oz.  take  J  of  a  pwt.  A/i5. 11^?^^^.  3^v. 
13w  From  ^  of  a  cwt.  take  T7^  of  a  Ib. 

Ans.  1  $r.  27  Ib.  6  oz.  lO,3^  ^- 

13.  From  3|  weeks,  take  ^  of  a  day,  and  -}  of  f  of  j-  of 
an  hour.  Ans.Qw.  &da.  12  ho.  \9min.  17^  sec. 


*  In  subtracting  mixed  numbers,  when  the  lower  fraction  is  greater  than 
the  upper  one,  you  may,  without  reducing  them  to  improper  fractions,  sub- 
tract the  numerator  of  the  lower  fraction  from  the  common  denominator, 
and  to  that  difference  add  the  upper  numerator,  carrying  one  to  the  unit's 
glace  of  the  lower  whole  number. 

Also,  a  fraction  may  be  subtracted  from  a  whole  number  by  taking  the 
numerator  of  the  fraction  from  its  denominator,  and  placing  the  remainder 
over  tile  (Denominator,  then  taking  oiw  from  the  whole  number. 


154  MULTIPLICATION,  DIVISION,  &Q. 

MULTIPLICATION  OF  VULGAR  FRACTIONS. 

RULE. 

-\C    v 

Reduce  whole  and  mixed  numbers  to  the  improper  frao 
tions,  mixed  fractions  to  simple  ones,  and  those  of  different 
integers  to  the  same  ;  then  multiply  all  the  numerators  to- 
gether for  a  new  numerator,  and  all  the  denominators  to- 
gether for  a  new  denominator. 

EXAMPLE?. 

1.  Multiply  |  by  -*-  Answers.   44=-J- 

2.  Multiply  f  by  ^  -  ££ 

3.  Multiply  5}  by  £ 

4.  Multiply  |  of  7  by  4-  3|i 

5.  Multiply  }Jf  by  Vv  M 
G.  Multiply  |  of  8  by  J  of  5 

7.  Multiply  7£  by  9} 

8.  Multiply  f  of  J  by  £  of  3}  f  £ 

9.  What  is  the  continued  product  of  J  of  f ,  7,  5J  aricL  J 
of  £  ?  u4/zs.  4  pV 

DIVISION  OF  VULGAR  FRACTIONS- 

RULE. 

Prepare  the  fractions  as  before  ;  then,  invert  the  divisor 
and  proceed  exactly  as  in  Multiplication  : — The  products 
will  be  the  quotient  required. 

EXAMPLES; 

4x5 

1.  Divide  -f  by  J  Thus, =|f  Ans. 

3x7 

3v  Divide  ^  by  J  ^?w?^r5.  1T% 

3.  Divide  £  of  |  by  £  £ 

4  What  is  the  quotient  of  17  by  f  ?  59^ 

5.  Divide  5  by  •& 

6.  Divide  i  of?,  of  f  by  |  of  J  3^ 

7.  Divide  4fr  by  '£  of  4  2.V 

8.  Divide  71  by  127  &V 

9.  Divide  52054-  bv  4  of  91 


RULE  OF  THREE  DIKECT,  INVERSE,  &,d.  155 

RULE  OP  THREE  DIRECT  IN  VULGAR 
FRACTIONS. 

RULE. 

Prepare  the  fractions  as  before,  then  state  your  questkw 
agreeable  to  the  Rules  already  laid  down  in  the  Rule  of 
Three  in  whole  numbers,  and  invert  the  first  term  in  the 
proportion  ;  then  multiply  all  the  three  terms  continually 
together,  and  the  product  will  be  the  answer,  in  the  same 
name  with  the  second  or  middle  term. 

EXAMPLES. 

1.  Iff  of  a  yard  cost  ~  of  a  pound,  what  will  ^  of  an  Ell 
English  cost  ? 

£yd.=f  off  of  £=!£  or  £  Ell  English. 
Ell  £.     EU.  s.  d.  yrs. 

As  £  :  -f  :  :  -£r  And  \  x  %  x  rV^fVi^-^0  3  H    An*- 

2.  If  f  of  a  yard  cost  J  of  a  pound,  what  will  40£  yards 
come  to  1  Ans.  £59  8s.  6±d. 

3.  If  50  bushels  of  wheat  cost  17f  Z.  what  is  it  per  bush- 
el? Ans.  7s.  Qd.  Iff  qrs. 

4.  If  a  pistareen  be  worth  14  J  pence,  what  are  100  pistar 
reens  worth  ?  Ans.  £6 

5.  A  merchant  sold  51  pieces   of  cloth,  each  containing 
24£  yards  at  9s.  ?id.  per  yard  ;  what  did  the  whole  amount 
to?  Ans.  £60  105.  Od.  3%  qrs. 

6.  A  person  having  f  of  a  vessel,  sells  f  of  his  share  for 
312Z.  ;  what  is  the  whole  vessel  worth  ?  Ans.  £780 

7.  If  |  of  a  ship  be  worth  f  of  her  cargo,  valued  at  8000?. 
what  is  the  whole  ship  and  cargo  worth  ? 

Ans.  £10031  14s. 


INVERSE  PROPORTION. 

RULE. 

Prepare  the  fractions  and  state  the  question  as  before^ 
then  invert  the  third  term,  and  multiply  all  the  three  terms 
.ogether.  the  product  will  be  tht?  answer; 


fcU-LE  OF  TH&EE  DMIBCC  IN  DECIMALS. 
EXAMPLE'S. 


1.  How  much  shalloon  that  is  J  yard  wid^t  will  line  5J,- 
yards  of  cloth  which  is  ]  J  yard  wide  ? 

Fic/5.  yr/5.  yds.  Yds. 

As  1J  :  5i  :  :  f     And  J  x  y  *f  =W—  *<>&  ^W5- 

2.  If  a  man  perform  a  journey  in  3}  days,  when  the  day 
is  12J  hours  long  ;   in  how  many  days  will  he  do  it  when 
the  day  is  but  9|  hours  ?  Ans.  47\4j  days. 

3.  If  13  men  in  11|  days,  mow  21  acres,  in  how  many 
days  will  8  men  do  the  same?  Ans.  18ff  days. 

4.  How  much   in   length  that   is  7±  inches  broad,  will 
make  a  square  foot  1  Ans.  20  inches. 

5.  If  25f  s.  will  pay  for  the  carriage  of  a  cwt.  145}  mites  ; 
how  far  may  6£  cwt.  he  carried  for  the  same  money  1 

Ans.  22^-  miles. 

6.  How  many  yards  of  baize  which  is   1}   yards  wide, 
will  line  18  J-  yards  of  camblet  J  yard  wide? 

Ans.  11  yds.  1  qr.  1J  net. 


RULE  OF  THREE  DIRECT  IN  DECIMALS. 

RULE. 

Reduce  your  fractions  to  decimals,  and  state  ytfur  ques- 
tion as  in  whole  numbers;  multiply  the  second  and  third  tfl* 
gether ;  divide  by  the  first,  and  the  quotient  will  b£  the  an- 
swer, &4N 

EXAMPLES* 

1.  If  I  of  a  yard  cost  TT¥  of  a  pound  ;  what  will  15 £  yards 
come  to  ?  i=,875T\=,583+  and  £=,75 

Yds.     £.          Yds.     £.  £.  s.  d.    qrs. 

As  ,875 :  ,583  :  :  15,75:  10,494=10  9  10  2,24  Ans. 

2.  If  1  pint  of  wine  cost  1,2s.  what  cost  12,5  hhds? 

Ans.  £378 
0.  If  4]  rafds  cost  3s.  44,d.  what  will  30|  yards  cost  t 


SIMPLE  INTEREST  BY  DECIMALS.  1*57 

4.  If  1,4  cwt.  of  sugar  cost  10  dols.  9  cts.,  what  will  9 
Civt.  3  qrs.  cost  at  the  same  rate  ? 

not.         $  cwt.       $ 

*  As  1,4  ::  10,09  :  :  9,75  :  70,269=$70,  26  cts.  9  m.+ 

5.  If  19  yards  cost  25,75  dollars,  what  will  4.35 J  yards 
come  to  1  Ans.  $590,  21  cts.  7f$  m. 

6.  If  345  yards  of  tape  cost  5. dols.  17  cents,  5  m.,  what 
will  one  yard  cost  1  Ans.  ,015=1^  cts. 

7.  If  a  man  lay  out  121  dollars  23  cts.  in  merchandise, 
and  thereby  gains  139,51  dollars,  how  much  will  he  gain  in 
laying  out  12  dollars  at  the  same  rate  ? 

Ans.  $3,91=$3,  91  cts. 

8.  How  many  yards  of  riband  can  I  buy  for  25J  dols.  if 
29J  yards  cost  4]  'dollars  1  Ans.  178£  yards. 

9.  If 178J  yds.  cost  25 J-  dollars,  what  cost  29f  yards  1 

Ans.  $4£ 

10.  If  1.6  cwt.  of  sugar  cost  12  dols.  12  cts.,  what  cost  3 
hhds.,  each  11  cwt.  3  qrs.  10,12  Ib.  ? 

Ans.  269,072  rfofc.=$2G9,  7  cts.  2m.+ 


SIMPLE  INTEREST  BY  DECIMALS. 

A  TABLE  OF  RATIOS. 


Rate  per  cent. 

Ratio.        \Rate  per  cent. 

Ratio. 

3 
4 
4 
5 

,03 
,04 
,045 
,05 

K\ 

? 

6i 

7 

,055 
,06 
,065 
,07 

Ratio  is  the  simple  interest  of  II.  for  one  year  ;  or  in  fe- 
deral money,  of  $1  for  one  year,  at  the  rate  percent,  agreed 
on, 

RULE. 

Multiply  the  principal,  ratio  and  time  continually  toge- 
ther, and  the  last  product  will  be  the  interest  required. 

EXAMPLES. 

1.  Required  the  interest  of  211  dols.  45  cts.  for  5  years> 
at  5  per  c.ent.  per  annum  ? 


1'08  SIMPLE"  •flSTTERlST'  fc¥  BECIMALV. 


$  cts. 

211,45  principal. 
,05  ratio. 


10,5725  interest  for  one  yeah 
5  multiply  by  the  time. 


52,8625     ylws.=g52,  86  cts.  2J  m. 

2.  What  is  the  interest  of  645?.  10s.  for  3  years,  at  &  p«i' 
cent,  per  annum  1 

<£645,5x06x3=116,190=£116  3s.  9d.  2,4  qrs.  Am. 

3.  What  is  the  interest  of  121/.  8s.  6d.  for  4£  years,  at 
6  per  cent,  per  annum  1          Ans.  £32  15s.  &d.  1,36  <£rs. 

4.  What  is  the  amount  of  536  dollars,  39  cents,  for  I£ 
years,  at  6  per  cent,  per  annum  1  Ans.  $584,6651. 

5.  Required  the  amount  of  648  dollars  50  cents  for  12J 
years,  at  5J  per  cent,  per  annum?       Ans.  $1103,26cte. 

CASE  II. 

The  amount,  time  and  ratio  given,  to  find  the  principal. 
RULE. — Multiply  the   ratio  by  the  time,    add  unity  to  the  product 
for  a  divisor,  by  which  sum  divide  the  amount,  and  the  quotient  will 
be  the  principal. 

EXAMPLES. 

1.  What  principal  will  amount  to  1235,975  dollars,  in  £» 
yenrs,  at  6  per  cent,  per  annum  1     $  $ 

,66*5+1=1,30,1235,975(950,75  Ans. 

2.  What  principal  will  amount  to  873/.  19s.  in  9  years^ 
at  6  per  cent,  per  annum  1  Ans.  £567  10s. 

3.  What  principal  will  amount  to  $626,  6  cents   in    12 
years,  at  7  per  cent.  1  Ans.  $340,25=$340,  25  cts. 

4.  What  principal  will   amount  to  956/.   10s.  4,125d.  in, 
BJ  years,  at  5J  per  eent.  ?  Ans.  £645  15s. 

CASE  III. 

The  amount,  principal  and  time  given,  to  find  the  ratio* 
RULE. — Subtract  the  principal  from  the  amount,  divide  the  re- 
mainder by  the  product  of  the  time  and  principal,  and  the  quotient 
will  be  the  ratio. 

EXAMPLES. 

1.  At  what  rate  per  cent,  will  950,75  dollars  amount  to 
#530,975  dollars  in  5  years  ? 


DIMPLE  INTEREST  BY  DECIMALS.  159 

From  the  amount     —     1235,975 
Take  the  principal  —       950,75 

950,75  x  5=-4753,75)2H572250(,06=6  per  cent. 

285,2250  Ans. 

Q.  At  what  rate  per  cent,  will  5677.  10s.  amount  to  8737. 
19s.  in  9  years  1  Ans.  6 per  cent. 

3.  At  what  rate  per  cent,  will  340  dols.  25  cts  amount  to 
626  dols.  6  cts.  in  12  years  1  Ans.  7  per  cent. 

4.  At  what  rate  per  cent,  will  6457.  15s.  amount  to  9567. 
10s.  4,125d.  in  8f  years  1  Ans.  5%  per  cent. 

CASE  IV. 
The  amount,  principal,  and  rate  per  cent,  given,  to  find 

the  time. 

RULE. — Subtract  the  principal  from  the  amount;  divide  the  re- 
mainder by  the  product  of  the  ratio  and  principal ;  and  the  quotient 
will  be  the  time. 

EXAMPLES. 

1.  In  what  time  will  950  dols.  75  cts.  amount  to  l23o 
dollars,  97,5  cents,  at  6  per  cent,  per  annum  t 
From  the  amount       §1235,975 
Take  the  principal     '    950,75 


050,75x06  =57,0450)285,2250(5  years, 
285,2250 


2.  In  what  time  will  5677.  10s.  amount  to  8737.  19s.  at 
(>  per  cent,  per  annum  1  Ans.  9  years. 

3.  In  what  time  will  340  dols.  25  cts.  amount  to  62)5 
dols.  6  cts.  at  7  per  cent  per  annum?         Ans.  12  years. 

4.  In  what  time  will  6457.   15s.   amount  to  9567.  10s, 
4?125d.  at  5|  per  ct.  per  annum  ?     Ans.  8,75=8J  years. 


TO  CALCULATE  INTEREST  FOR  DAYS. 

RULE. — Multiply  the  principal  by  the  given  number  of  days,  and 
that  product  by  the  ratio  ;  divide  the  last  product  by  365  (the  num- 
ber of  days  in  a  year)  and  it  will  give  the  interest  required. 
EXAMPLES. 

J .  What  is  the  interest  of  3007. 10s.  for  146  days,  at  6  rfr.  ct.? 


160 


SIMPLE    INTEREST    BY    DECIMALS, 


360,5  xl  46  x, 06      £        £  s.  d.  qrs. 

-—=8652=8  13  0  1,9 

365 

2.  What  is  the  interest  of  640  dols.  60  cts.  for  100  days, 
at  6  per  cent,  per  annum  ?  Ans.  $10,  53.cfo.-f 

3.  Required  the  interest  of  250Z.  17$.  for  120  days,  at  5 
per  cent,  per  annum  ?  Ans.  £4,1235=47.  2s.  5J«£  + 

4.  Required  the  interest  of  481  dollars  75   cents,  for  25 
days,  at  7  per  cent,  per  annum  ?     Ans.  $2,  30  cts.  9m.  + 


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SIMPLE  INTEREST  BY  DECIMALS, 


16-i 


When  interest  is  to  be  calculated  on  cash  accounts,  &cu 
where  partial  pay  meats  are  made ;  multiply  the  several 
balances  into  the  days  they  are  at  interest,  then  multiply 
the  sum  of  these  products  by  the  rate  oa  the  dollar,  and  di- 
vide the  last  prti'jdot  by  ->135,  ;n<J  you  will  have  the  whole 
interest  due  0,1  the  liecn;2::!,  ebc. 

^. 

Lent   Peter   Trusty,   wr  bill   on   Jc  A    1st  of 

June,  1800,  V')CO  dollars,  of  which  1  received  bac.k  the  19th 
of  August,  400  dollars  ;  o;i  the  lath  of  October,  60t) 
dollars;  on  the  llth  of  Deqsniber,  400  iJoJl.^rs  ;  on  the 
17th  of  February,  1801,  SQO  dollars  ;  a<*i  or-  tht  1st  of 
Jur.i.'  400  dollars  ;  how  much  interest  is  due  oa  the  bill, 
reckoning  ftt  6  per  cent.  1 


1800.  dels. 
June  1,  .Principal  per  bill,  2(fGC« 
August  19,  Received  in  part,  40b 

a  ays. 
7'J 

57 
57 
6S 
104 

"oro  ducts* 
158000 

91200 
57GOO 
40800 
41GOO 

Balance,  160(1 
October  15,  Roccivod  in  p«rt,  600 

Balance,  1000 
December  11,  "Received  m  part,  400 

1801.  Balance,  600 
February  17,  Received  in  part,  200 

Balance,  400 
June  1,  Rec'd  in  full  of  principal,  400 

Then  388600 


388600 


,06  Ratio. 

$  cts.  m. 

365)23316,00(63,879  Ans.  =  63  87  9-f 
•)llv^vi!'jj  Ri.i?  f,»r  Computing-  interest  on  any  note, 
or  olbl i gallon,  when  the;-'  -•*.  f/ .  .pnynients  in  pnrt,  or  endorse- 
ments, v.  Superior  Court  of  the  i 
of  Connecticut,  in  1784, 

o  -2 


t62  SIMPLE  INTEREST  BY  DECIMALS, 

RULE. 

""  Compute  the  interest  to  the  time  of  the  first  payment ; 
if  that  be  one  year  or  more  from  the  time  the  interest  com- 
menced, add  it  to  the  principal,  and  deduct  the  payment 
from  the  sum  total.  If  there  be  after  payments  made, 
compute  the  interest  on  the  balance  due  to  the  next  pay- 
ment, and  then  deduct  the  payment  as  above,  and  in  like 
manner  from  one  payment  to  another,  till  all  the  payments 
are  absorbed  ;  provided  the  time  between  one  payment  and 
another  be  one  year  or  more.  But  if  any  payment  be  made 
before  one  year's  interest  hath  accrued,  then  compute  the 
interest  on  the  principal  sum  due  on  the  obligation  for  one 
year,  add  it  to  the  principal,  and  compute  the  interest  on 
the  sum  paid,  from  the  time  it  \vas  paid,  up  to  the  end  of 
the  year  :  add  it  to  the  sum  paid,  and  deduct  that  sum  from 
the  principal  and  interest  added  as  above.* 

"  If  any  payments  be  made  of  a  less  sum  than  the  interest 
arisen  at  the  time  of  such  payment,  no  interest  is  to  be  com- 
puted but  only  on  the  principal  sum  for  any  period." 

Kirby's  Reports,  page  49. 

EXAMPLES. 

A  bond,  or  note,  dated  January  4th,  1797,  was  given  for 
1000   dollars,  interest  at  6  per  cent,  and  there  were  pay- 
ments endorsed  upon  it  as  follows,  viz.  § 
1st  payment  February  19,  1798,                               200 
2d  payment  June  29/1799,                                        500 
3d  payment  November  14, 1799,                              260 
I  demand  how  much  remains  due  on  said  note  the  24th 
of  December,  1800? 

1000,00  dated  January  4,  1797. 

67,50  interest  to  February  19,  1 798=13 \  months. 

1067,50  amount.  [Carried  up.] 


*  If  a  year  does  not  extend  beyond  the  time  of  final  settlement  j  but  if  it 
floesj  then  find  the  amount  of  the  principal  sum  due  on  the  obligation,  up  to 
the  time  of  settlement,  and  likewise  find  the  amount  of  the  sum  paid,  from  the 
lime  it  was  paid,  up  to  the  time  of  the  final  settlement,  and  deduct  this 
amount  from  the  amount  of  the  principal.  But  if  there  be  several  payments 
jnade  within  the  said  time,  find  the  amount  of  the  several  payments,  from 
the  time  they  were  paid,  to  the  time  of  settlement,  und  deduct'their  amount 
from,  the  amount  tn  the  principal. 


SIMPLE  INTEREST  BY  DECIMALS.  163 

1067,50     amount.  [Brought  up. 

200,00     first  payment  deducted. 

867,50     balance  due,  Feb.  19,  1798. 
70,845  interest  to  June  29,  1799=161  months. 

938,345  amount. 

500,000  second  payment  deducted. 

438,345  balance  due  June  29,  1799. 
26,30     interest  for  one  year. 

464,615  amount  for  one  year. 

269,7  :U  amount  of  third  payment  for  7|  months.* 

194,895  balance  due  June  29,  1800.  mo.  da. 

5,687  interest  to  December  24,  1800.  5     25 


200,579  balance  due  on  the  Note,  Dec.  24,  1800. 
RULE  II. 

E-stablished  by  the  Courts  of  Law  in  Massachusetts  for 
computing  Interest  on  notes.  Sfc.  on  which  partial  pay- 
ments  have  been  endorsed. 

"  Compute  the  interest  on  the  principal  sum,  from  the 
time  when  the  interest  commenced  to  the  first  time  when 
a  payment  was  made,  which  exceeds  either  alone  or  in  con- 
junction with  the  preceding-  payment  (if  any)  the  interest  at 
that  time  due  :  add  that  interest  to  the  principal,  and  from 
the  sum  subtract  the  payment  made  at  that  time,  together 
with 'the  preceding  payments  (if  any)  and  the  remainder 
forms  a  new  principal  ;  on  which  compute  mid  subtract 
the  payments  as  upon  the  first  principal,  and  proceed  ia 
this  manner  to  the  time  of  final  settlement." 


$   cla. 

*260,00  third  payment  with  its  interest  from  the  time  it  was  paid,  up  ia 
9,75      the  end  of  the  year,  or  from  Nov.  14,  1799,  to  June  29,  1800, 
*——  —      which  is  7  and  1-2  months. 
269,75  amount, 


164  'SIMPLE  INTEREST  BY  DEClAIALS. 

Let  the  foregoing  example  be  solved  by  this  Rule. 
A  note  for  1000  dols.  dated  Jan.  4,  1797,  at  t>  per  cent. 
1st  payment  February  19,  1798,  $200 

2d  payment  Ju«ie  '29,  1799,  *500 

3d  payment  November  14,  1799,  260 

How  much  rcu'.a^'.s  due  o,»  fcal-J  iiOte  th*j, ?  5<J,  ^  .Decem- 
ber, \-  r.ts. 
Fnncii>ti5,  Jaiii!;:;y  -i,  I                                            1000,00 
Interest  to  February  ki-",  1798.  (U>-£  i?z«?.)                67,60 

Amount,     1007,50 
ary  19,  17  ^00,00 


Kfr,,-                                        .|>al,  867,50 

Interest  to  Ju                  ;;-y,  (itfj  wr.)  70,84 

Amount,  938,34 

Paid  Jiuic-iO,  17  500,00 


Reirii,. 


:  U 


November    14,  17f  ), 


•.-il,  188,20 

rc-rt  ;••-  $ 


Balance  due  on  said  note,  Dec.  24,  1800,          200,90 

$     cts. 

The  balance  by  Rule  I.  200,579 
'  Rule  II.  200,990 

DrflViwce,      0,411 
Aiv  le  MI  RuVj'U. 

A  hor.f?  or  i;ary  1,  "1800,  v,-j«v   pven  for 


--.  ^  »«*  ;: 

-^-  ^  '**• 

s»j.  ;•               Way  T  '-^00.  40,00 

2d  payment  November  14,  1800  8,00 


COMPOUND  INTEREST  BY  DECIMALS.  165 

3d  payment  April  1,  1801,  12,00 
4th  payment  May  1,  1801,  30,00 
How  much  remains  due  on  said  note  the  16th  of  Sep- 
tember, 1801  ?  $  cts. 
Principal  dated  February  1,  1800,  500,00 
Interest  to  May  .1,  1800,  (3  mo.)  7,50 

Amount  507  50 
Paid  May  1,  1800,  a  sum  exceeding  the  interest       40,00 

'  New  principal,  May  1,  1800,  467,50 

Interest  to  May  1,  1801,  (1  year,)  28,05 

Amount     495,55 
Paid  Nov.  4,  1800,  a  sum  less  than  the 

interest  then  due,  8,00 

Paid  April  1,  1801,     do.     do.  12,00 

Paid  May  1, 1801,  a  sum  greater,      30,00 

50,00 


New  principal  May  1,  1801,  445,55 

Interest  to  Sept.  16,  1801,  (4J  mo.)  10,92 

Balance  due  on  the  note,  Sept.  16,  1801,         $455,57 
OJr^The  payments  being  applied  according  to  this  Rule, 
keep  down  the  interest,  and  no   part   of  the   interest   ever 
forms  apart  of  the  principal  carrying  interest. 


COMPOUND  INTEREST  BY  DECIMALS.  . 

RULE. — Multiply  the  given  principal  continually  by  the 
amount  of  one  pound,  or  one  dollar,  for  one  year,  at  the 
rate  per  cent,  given,  until  the  number  of  multiplications  are 
equal  to  the  given  number  of  years,  and  the  product  will 
be  the  amount  required. 

Or,  In  Table  I,  Appendix,  find  the  amount  of  one  dollar, 
or  one  pound,  for  the  given  number  of  years,  which  multiply 
by  the  given  principal,  and  it  will  a'ive  the  amount  as  before* 


J6,G  INVOLUTION. 

EXAMPLES. 

1.  What  will  400Z.  amount  to  in  4  years,  at  6  per  cent, 
par  annum,  compound  interest  ? 

400x1,06     1,00 x  1,06 x  l,06=£504,99+oj 

[£504  19s.  9d.  2,75  ^.-f  AM. 
The  same  by  Table  I. 
Tabular  amount  of  £1  =  1,26-247 
Multiply  by  the  principal  400 

Whole  amount=£504,98800 

2.  Required  the  amount  of  425  dols.  75  cts.  for  3  years, 
at  6  per  cent,  compound  interest?      Ans.  $507, 7 J  cts. -f 

3.  What  is  the  compound  interest  of  555  dols.   for    14 
years  at  5  per  cent.?     By  Table  I.       Ans.  543,86  cts.+ 

4.  What  will  50  dollars  amount  to  in  20  years,  at  6  per 
cent,  compound  interest?  Ans.  §160,  35  cts.  6£ra. 

INVOLUTION, 

IS  the  multiplying  any  number  with  itself,  and  that  pro- 
duct by  the  former  multiplier ;  and  so  on  ;  and  the  several 
products  which  arise  are  called  powers. 

The  number  denoting  the  height  of  the  power,  is  called 
the  index  or  exponent  of  that  power. 

. 

EXAMPLES. 

What  is  the  5th  power  of  8  ? 
8  the  root  or  1st  power, 

8 

64  =2d  power,  or  square,. 

8 

512  =3d  power,  or  cube. 

8 

<**r . 

4096  =4th  power,  or  biquadrate. 

8 


2768  —5th  power,  or  sitrsolkl. 


EVOLUTION,  Oft  E'XTRA6Tlfc>lV  Or  lldOl'S.  16? 

What  is  the  square  of  17,1  ?  Ans.  292,41 

What  is  the  square  of  ,085  ?  -4ns.  ,007225 

What  is  the  cube  of  25,4  1  Ans.  16387,004 

What  is  the  biquadrate  of  12  1  4ns.  20730 

What  is  the  square  of  7J  ?  -4ns. 


EVOLUTION,  OR  EXTRACTION  OF  ROOTS. 

WHEN  the  root  of  any  power  is  required,  the  business 
of  finding  it  is  called  the  Extraction  of  the  Root. 

The  root  is  that  number,  which  by  a  continued  multipli- 
cation into  itself,  produces  the  given  power. 

Although  there  is  no  number  but  what  will  produce  a 
perfect  power  by  involution,  yet  there  are  many  numbers  of 
which  precise  roots  can  never  be  determined.  But,  by  the 
help  of  decimals,  we  can  approximate  towards  the  root  to 
any  assigned  degree  of  exactness. 

The  roota  which  approximate  are  called  surd  roots,  and 
those  which  are  perfectly  accurate  are  called  rational  roots, 

A  Table  of  the  Squares  and  Cubes  of  the  nine  digits. 


Roots. 

M 

|2 

1    3 

4 
16  | 

5 

1      6 

7 

| 

Squares. 

u 

L4 

9 

25 

|    36 

49 

1 

Cubes. 

11 

18 

27 

64  | 

125 

|216| 

343 

L 

EXTRACTION  OF  THE  SQUARE  ROOT. 

Any  number  multiplied  into  itself  produces  a  square. 

To  extract  the  square  root,  is  only  to  find  a  number, 
which  being  multiplied  into  itself  shall  produce  the  given 
number. 

RULE. — 1.  Distinguish  the  given  number  into  periods  of 
two  figures  each,  by  putting  a  point  over  the  place  of  units, 
another  over  the  place  of  hundreds,  and  so  on  ;  and  if 
there  are  decimals,  point  them  in  the  same  manner,  from 
units  towards  the  right  band  ;  which  points  show  the  num- 
ber of  figures  the  root  will  consist  of. 

2.  Find  the  greatest  square  number  in  the  first,  or  left 
hand  period,  place  the  root  of  it  at  the  right  hand  of  th^ 


168  TiVOLUT'lOtt,  Oil  EXTRACTION  OF 

given  number,  (after  the  manner  of  a  quotient  in  division,) 
for  the  first  figure  of  the  root,  and  the  square  number  un- 
der the  period,  and  subtract  it  therefrom,  and  to  the  re- 
mainder bring  down  the  next  period,  for  a  dividend. 

3.  Place  the  double  of  the    root,   already   found,  on  the 
left  hand  of  the  dividend,  for  a  divisor. 

4.  Place  such  a  figure  at   the  right  hand  of  the  divisor, 
and  also  the  same  figure  in  the   root,    as   when    multiplied 
into  the  whole  (increased  divisor)  the  product  shall  be  equal 
to,  or  the  next  less  than  the  dividend,  and    it   will  be  the- 
second  figure  in  the  root. 

5.  Subtract  the   product    from  the  dividend,    and  to  the 
remainder  join  the  next  period  for  a  new  dividend. 

6.  Double  the  figures   already  found    in  the   root,  for  a 
new  divisor,  and  from  these  iind  the  next  figure  in  the  root 
as  last  directed,   and  continue  the  operation  in  the  same 
manner  till  you  have  brought  down  all  the  periods. 

Or,  to  facilitate  the  foregoing  Rule,  when  you  have 
brought  down  a  period,  and  formed  a  dividend  in  order  to 
find  a  new  figure  in  the  root,  you  may  divide  said  dividend 
(omitting  the  right  hand  figure  thereof)  by  double  the  root 
already  found,  and  the  quotient  will  commonly  be  the 
figures  sought,  or  being  made  less  one  or  two,  will  generally 
give  the  next  figure  in  the  quotient. 

EXAMPLES. 

1.   Required  the  square  root  of  141225,64. 

141225,64(375,8  the  root  exactly  without   a  remainder  ; 

9  but  when  the  periods  belonging  to  any 

given  number  are  exhausted,  and  still 

67)512  leave  a   remainder,  the  operation  may 

469  be  continued  at  pleasure,  by  annexing 

periods  of  ciphers,  &c. 
745)4325 
3725 


7508)60064 
60064 


0  remains, 


^.   What  is  the  square  root  of  1296  '?  ANSWERS.  36 

3.  Of  5<J344?  23,8 

4.  Of  5499025?  2345 

5.  Of  36372961?  6031 

6.  Of  181,2?  13,57 -t 

7.  Of  0712,693801)?  98,553 

8.  Of  0,45369?  ,673+ 

9.  Of  ,00291(5?  ,054 
10.  Of  —                            45?  6,TOS+ 


TO  EXTRACT  THE  SQUARE  HOOT  OF  VUL- 
GAR  FRACTIONS. 

RULE. 

Reduce  the  fraction  to  its  lowest  terms  for  this  and  all 
other  roots  ;  then 

1.  Extract  the  root  of  the  numerator  for  a  new  numera- 
tor, and  the  root  oflhe  denominator,  for  a  new  denominator. 

2.  If  the  fraction  be  a  surd,  reduce  it  <o  a  decimal,  and 
extract  its  root. 

EXAMPLE^. 

1.  "What  is  tiie  square  rootof  ,!V.  ?  ANSWERS.  £ 


? 


2.  What  is  the  square  root  of  f ,f 2 . 

3.  What  is  the  square  root  of  -f -}->  ?  J 
4..  What  is  the  square  root  of  20}  ?  4< 

5.  What  is  the  square  root  of  243,^  '  I.5J- 

SURDS. 

6.  What  is  the  square  root  of  Jf  ?  9128 -f 

7.  What  is  the  square  root  of  41  ?  ,7745 -f 
8-.  Required  the  square  root  of  36£  ?  6,0207-^- 

APPLICATION  AND~USE  OE  THE  SQUARE 

ROOT. 

PROBLEM  I. — A  certain  general  has  an  army  of  5184 

men ;  how  many  must  he   place  in  rank  and  file,  to  form 
them  into  o.  square  ? 


170  EVOLVTiON,  OR  EXTRACTION  OF  KOD'i>. 

RULE.  —  Extract  the  square  root  of  the  given  number. 

V5184=7£  Am. 

PROB.  II.  A  certain  square  pavement  contains  20736 
square  stones,  all  of  the  same  size;  I  demand  how  many 
are  contained  in  one  of  its  sides?  \/20736—  144  Ans. 

PROB.  III.  To  find  a  mean  proportional  between  two 
numbers. 

RULE.  —  Multiply  the  given  numbers  together  and  extract 
the  square  root  .of  the  product. 


What  is  the  mean  proportional  between  18  and  7:2  'I 

72  x  18=1296,  and  V  1296=36  Ans. 

PROF..  IV.  To  form  any  body  of  soldiers  so  that  they  may 
be  double,  triple  &c.  as  many  in  rank  as  in  file. 

RULE.  —  Extract  the  square  root  of  1-2,  1-3,  &c.  of  the 
given  number  of  men,  and  that  will  be  the  number  of  men 
in  file,  which  double,  triple,  &c.  and  the  product  will  be  tife 
number  in  rank. 

EXAMPLES. 

Let  1312*2  men  be  so  formed,  as  that  the  number  in  rank 
may  be  double  the  number  in  file. 

13122-^2=6561,  and  ^6561=81  in  file,  and  81x2 
=162  in  rank. 

PROB.  V.  Admit  10  hhds.  of  water  are  discharged 
through  a  leaden  pipe  of  2£  inches  in  diameter,  in  a  cer- 
tain time  ;  I  demand  what  the  diameter  of  another  pipe 
must  be  to  discharge  four  times  as  much  water  in  the  same 
time. 

RULE.  —  Square  the  given  diameter,  and  multiply  said 
square  by  the  given  proportion,  and  the  square  root  of  the 
product  is  the  answer. 

2£=2,5,  and  2,5x2,5=6,25  square. 

4  given  proportion. 

^  25,00=5  inch,  diam,  Ans. 


E  VO.L U TIQX ,  OR  E XT R A (  T 1 0 X  Q F  II Q OT §..  171 

PRQB.  VI.  The  sum  of  any  two  numbers,  and  their  pro- 
ducts being  given,  to  find  each  number. 

RULE-. — From  the  square  of  their  sum,  subtract  4  times  their  pro- 
duct, and  extract  the  square  root  of  the  remainder,  which  will  be  the 
difference  of  the  two  numbers;  then  half  the  said  difference  added  to 
half  the  sum,  gives  the  greater  of  the  two  numbers,  and  the  said  ha!i' 
difference  subtracted  from  the  half  sum.,  gives  the  lessor  imr-ibor. 

EXAMPLES. 

The  sum  of  two  numbers  is  43,  and  their  product  is  442 ; 
what  are  those  two  numbers  ? 

The  sum  of  the  numb.  43  X  43=^1849  square  of  do. 

The  product  of  do.       442  x    4=^1768  4  times  the  pro, 
Then  to  the  -?r  sum  of  21 ,5  [.numb. 

•f  and—  4,5  A/81=9  diff.  of  the 


Greatest  n  anbcr,        '20,0  )  4£  the  %  dift'. 

>  Answers^ 

Least  n  limber,  17,0  J 


EXTRACTION  OF  THE  CUBE  ROOT. 

A  cube  is  any  number  multiplied  by  its  square. 

To  extract  the  cube  root,  is  to  find  a  number,  which,  be- 
ing multiplied  into  its  square,  shall  produce  the  given  num- 
ber. 

RULE. 

1.  Separate  the  given  number  into  periods  of  three  figures 
each,  by  putting  a    point  over  the  unit  figure,  and  c very- 
third  figure  from  the  place  of  units  to  the  left,  and  if  there 
be  decimals,  to  the  right. 

2.  Find  the   greatest  cube  in   the   left  hand  period,  and 
place  its  root  in  the  quotient. 

3.  Subtract  the  cube  thus  found,  from  the  said   period, 
and  to  the  remainder  bring  down  the  next  period,  calling 
this  the  dividend. 

4.  Multiply  the  square  of  the  quotient  by  300,  calling  it 
the  divisor. 


f.  \  OLUTHJX.  OK  KXTRAi    PlOJi    6F   ROC T^ 

5.  rJeek  how  often  the  divisor  may  be  had  in  the  divi- 
dend, and  place  the  result  in  the  quotient ;  then  multiply 
the  divisor  by  this  last  quotient  figure,  placing  the  product 
under  the  dividend. 

G.  Multiply  the  former  quotient  figure,  or  figures,  by  the 
square  of  the  last  quotient  figure,  and  that  product  by  30, 
and  place  the  product  under  the  last ;  then  under  these  two 
products  place  the  cube  of  the  last  quotient  figure,  and  add 
them  together,  calling  their  sum  the  subtrahend. 

7.  Subtract  the  subtrahend  from  the  dividend,  and  to  the 
remainder  bring  down  the  next  period  for  a  new  dividend  ; 
with  which  proceed  in  the  same  manner,  till  the  whole  be 
finished. 

NOTE. — If  the  subtrahend  (found  by  the  foregoing  rule) 
happens  to  be  greater  than  the  dividend,  and  co;A  "equently 
cannot  be  subtracted  therefrom,  you  must  make  the  last 
quotient  figure  one  less;  with  which  find  a  new  subtrahend, 
(by  the  rule  foregoing,)  and  so  on  until  you  can  subtract 
the  subtrahend  from  the  dividend. 

EXAMPLES. 

I.  Required  the  cube  root  of  18399,744. 

18399,744(26,4  Root.  Ans*. 
8 


2x2r,,4x  300=1200)10399  first  dividend, 

.      7200 

6  x  6^36  X  2=72  x  30=2160 
6x6x6=  216 

9576  1st  subtrahend, 
•26  X 26=676  x  300=202800)823744  2d  dividend. 

811200 

4x4=-16x26rr=416x30r=:  12480 
4X4X4=          64 

"-W44  2d  .suM 


£  V  <4L  L'.T  ID  r\  ,  OR  E  X  T  R  A  (  .  1"  1  '.  •• 

NOTE,  —  The  foregoing  example  gives  a  perfect  root  ; 
and  if,  when  all  the  periods  are  exhausted,  there  happens 
to  be  a  remainder,  you  may  annex  periods  of  ciphers,  and 
cqn  tin  ue  the  operation  as  far  as  you  think  it  necessary. 


2.  What  is  the  cube  root  of  '205379  ?  59 

3.  Of  614125?  85 
4*.  Of  41421736  ?  346 

5.  Of  146363.183  ?  52,7 

6.  Of  29,508381  ?  3,09+ 

7.  Of  80,763  ?  4,32-1. 

8.  Of  -           ,162771336?  ,546* 

9.  Of  ,000684134?  ,088+ 
10.  Of  ---      32261n327232?  4968 

RULE.  —  1.  Find  by  trial,  a  cube  near  to  the  givon  number,  and  call 
it  the  supposed  cube. 

2.  Then,  as  twice  the  supposed  cube,  added  to  the  given  number,  is 
lo  twice  the  given  number  added  to  tlis  supposed  cube,  so  is  the  root 
of  the  supposed  cube,  to  the  true  root,  or  an  approximation  to  it. 

3.  By  taking  the  cube  of  the  root  thus  found,  for  the  supposed  cube. 
and  repeating  the  operation,  the  root  will  be  had  to  a  greater  degrou 
of  exactness. 

EXAMPLES. 

1.  Let  it  be  required  to  extract  the  cube  root  of  2. 
Assume  1,3  as  the  root  of  the  nearest  cube  ;  then  —  1.3  X 
1  ,3  X  1,3—  2,197=supposed  cube. 
Then,  2,197  2,000  given  number. 

2  2 


4,394  4,000 

2,000  2,197 


As  6,394      :      6,197      :     :      1,3     :     1,2599  root, 
which  is  true  to  the  last  place  of  decimals  ;  but  might  by  re^ 
•peating  the  operation,  be  brought  to  greater  exactness. 
2.  Wliat  is  the  cube  root  of  584,277056  ? 

Ans.  8,36. 
p  2 


174  K-VOLUTI.O:N,  OK    EXTRACTION   OF  ROOTS. 

3.    Required  the  cube  root  of  729001101  ? 

Ans.  900,0004. 

QUESTIONS, 

Showing  the  use  of  the  Cube  Root. 

1.  The  statute  bushel  contains  2150,425  cubic  or  solid 
inches.    I  demand  the  side  of  a  cubic  box,  which  shall  con- 
tain that  quantity  ? 

Z/  21 50,425=  12,907  inch.  Ans. 

NOTE. — The  solid  contents  of  similar  figures  are  in  pro- 
portion to  each  other,  as  the  cubes  of  their  similar  sides  or 
diameters. 

2.  If  a  bullet  3  inches  diameter  weigh  4  Ib.  what  will  a 
bullet  of  the  same  metal  weigh,  whose  diameter  is  6  in- 
ches ? 

3x3x3=27  6x6x6=216.  As  27  :  4  Ib.  :  :  216: 
32  Ib.  Ans. 

3.  If  a  solid  globe  of  silver,  of  3  inches  diameter,  be 
worth  150  dollars ;  what  is  the  value  of  another  globe  of 
silver,  whose  diameter  is  six  inches? 

3x3x3=27  6x6x6=216,  As  27  :  150  :  :  216  : 
$1200.  Ans. 

The  side  of  a  cube  being  given,  to  find  the  side  of  that 
cube  which  shall  be  double,  triple,  &c.  in  quantity  to  the 
given  cube. 

RULE. — Cube  your  given  side,  and  multiply  by  the  given  propor- 
tion between  the  given  and  required  cube,  and  the  cube  root  of  the 
product  will  be  the  side  sought. 

EXAMPLES. 

4.  If  a  cube  of  silver,  whose  side  is  two  inches,  be  worth 
20  dollars  ;  I  demand  the  side  of  a  cube  of  like  silver  whose 
value  shall  be  8  times  as  much  ? 

2  x  2  x  2—8,  and  8  X  8=64  ^/64=4  inches.     Ans. 

5.  There  is  a  cubical  vessel,  whose  side  is  4  feet ;  I  de- 
mand the  side  of  another  cubical  vessel,  which  shall  con- 
tain 4  times  as  much  ? 

4  x  4  x  4=64,  and  64  x  4=256  v/ 256=6, 349 +/*.  Am. 
6\  A  cooper  fiaving  a  ca^sk  40  inches  long,  and  32  in- 


EVOLUTION,  OR  EXTRACTION  OF  ROOTS.      175 

ches  at  the  bung  diameter,  is  ordered  to  make  another  cask 
of  the  same  shape,  but  to  hold  just  twice  as  much  ;  what 
will  be  the  bung  diameter  and  length  of  the  new  cask  ? 

40  X  40  x  40  X  2=128000  'then  V  1 28000—50,3  -f  kng? \ . 


32  X  32  x  32  x  2=65536  and  -^65536—40,3+  bung  diajfi. 

A  General  Rule  for  extracting  the  Roots  nf  all  Powers. 
RULE. 

1.  Prepare  the  given  number  for  extraction,  by  pointing 
off  from  the  unit's  place,  as  the  required  root  directs. 

2.  Find  the  first  figure  of  the  root  by  trial,  and  subtract 
its  power  from  the  left  hand  period  of  the  given  number. 

3.  To  the  remainder*  bring  down  the  first  figure  in  the 
next  period,  and  call  it  the  dividend. 

4.  Involve  the  root  to  the  next  inferior  power  to   that 
which  is  given,  and  multiply  it  by  the  number  denoting  the 
given  power,  for  a  divisor. 

5.  Find  how  many  times  the  divisor  may  be  had  in  the 
dividend,  and  the  quotient  will  be  another  figure   of  the 
root. 

6.  Involve  the  whole  root  to  the  given  power,  an^sub- 
tract  it  (always)  from  as  many  periods  of  the  given  number 
as  you  have  found  figures  in  the  root. 

7.  Bring  down  the  first  figure  of  the  next  period  to  the 
remainder  for  a  new  dividend,  to  which  find  a  new  divisor 
as  before,  and  in  like  manner  proceed  till  the  whole   ho 
finished. 

NOTE. — When  the  number  to  be  subtracted  is  greater 
than  those  periods  from  which  it  is  to  be  taken,  the  last 
quotient  figure  must  be  taken  less,  &c. 

EXAMPLES. 

1.  Required  the  cube  root  of  135796.744  by  the  above 
general  method. 


176       EVOLUTION,  OR  EXTRACTION  OF  ROOTS. 

135796744(51,4  the  root. 
125=lst  subtrahend. 


5)107  dividend. 

132651=2d  subtrahend. 
7803)  31457=2d  dividend. 

135796744=3d  subtrahend. 


5  X  5  x  3=75  first  divisor. 
51  x  51  x  51=132651  second  subtrahend. 
51  X  51  X  3=7803  second  divisor. 
514x514x514=135796744  3d  subtrahend. 

2.  Required  the  sursolid  or  5th  root  of  6436343. 

0436343(23  root. 
32 

2x2x2x2x 5=80)323  dividend. 

23  x  23  x  23  x  23  x  23=6436343  subtrahend. 

NOTE. — The  roots  of  most  powers  may  be  found  by  the 
square  and  cube  roots  only ;  therefore,  when  any  even 
power  is  given,  the  easiest  method  will  be  (especially  in  a 
very  high  power)  to  extract  the  square  root  of  it,  which  re- 
duces it  to  half  the  given  power,  then  the  square  root  of 
that  power  reduces  it  to  half  the  same  power;  and  so  an, 
till  you  come  to  a  square  or  a  cube. 

For  example :  suppose  a  12th  power  be  given  ;  the  square 
root  of  that  reduces  it  to  a  6th  power :  and  the  square  root 
of  a  6th  power  to  a  cube. 

EXAMPLES. 

3.  What  is  the  biquadrate,  or  4th  root  of  19987173376? 

Ans.  376. 

4.  Extract  the  square,  cubed,  or  6th  root  of  12S30590 
464.  Ans.  4& 

5.  Extract  the  square,  biquadrate,  or  8th  root  of  72138 
95789338336.  Arts.  96. 


ALLIGATION.  177 

V 

ALLIGATION, 

IS  the  method  of  mixing  several  simples  of  different  qua- 
lities, so  that  the  composition  may  be  of  a  mean  or  middle 
quality:  It  consists  of  two  kinds,  viz.  Alligation  Medial, 
and  Alligation  Alternate. 

ALLIGATION  MEDIAL, 

Is  when  the  quantities  arid  prices  of  several  things  afe 
given,  to  find  the  mean  price  of  the  mixture  composed 
of  those  materials. 

RULE. 

As  the  whole  composition  :  is  to  the  whole  value  :  :  so 
is  any  part  of  the  composition  :  to  its  mean  price. 

EXAMPLES. 

1.  A  farmer  mixed  15  bushels  of  rye,  at  64  cents  a  bush- 
el, 18  bushels  of  Indian  corn,  at  55  cts.  a  bushel,  and  21 
bushels  of  oats,  at  28   cts.    a  bushel  ;     I   demand  what  a 
bushel  of  this  mixture  is  worth  1 

bit.       cts.  $cts.          hn.       $  cts.        bit. 
15  at  64-9,60    As  54  :  25,38  :   :  1 
18       55=9,90  1 

21       28=5,88  --  cts. 

54)25,38(,47  Ans. 
54  25,38 

2.  If  20  bushels  of  wheat  at  I  dol.    35   cts.    per  bushel 
be  mixed  with  10   bushels  of  rye   at  90  cents  per  bushel, 
what  will  a  bushel  of  this  mixture  be  worth  1 


3.  A  tobacconist  mixed  30  Ib.   of  .tobacco,   at  Is.  (jd. 
per  Ib.  12  Ib.   at  2s.    a  pound,  with  12   Ih.  at  Is,  10d.  per 
Ib.  ;  what  is  the  price  of  a  pound  of  this  mixture  1 

Ans.   Is.  S(L 

4.  A  grocer  mixed  2  C.  of  sugar  at  56s?.  per  C.  and   1 
C.  at  43s.  per  C.  and  2  C.  at  50s.  per  C.  together  ;  I  de- 
mand the  price  of  3  cwt.  of  this  mixture  ?  Ans.  £7  13s. 

5.  A    wine  merchant   mixes    15   Dillons  of  wine  at  4s. 
2d.  per  gallon,  with  24  gallons    at  6s.  8d.  and  20    gallons 
at  6?.  3d.  ;  what  is  a  gallon  of  this  composition  worth  ? 

Ans.  5.s\  1(V/..  24-3.  ^r^ 


178  ALLIGATION'  ALTERNATE. 

*f>.  A  grocer  hath  several  sorts  of  sugar,  viz.  one  sort  at 
8  dols.  per  cwt.  another  sort  at  9  dols.  per  cwt.  a  third  sort 
at  10  dols.  per  cwt.  and  a  fourth  sort  at  12  dols.  per  cwt. 
nnd  he  would  mix  an  equal  quantity  of  each  together;  1 
demand  the  price  of  3^  cwt.  of  this  mixture  ? 

Am.  §34  \Zcts.5m. 

7.  A. goldsmith  melted  together  5  Ib.   of  silver  bullion, 
of  8  oz.  fine,  10  Ib.  of  7  oz.   fine,  and  15  Ib.  of  6  oz.  fine ; 
pray  what  is  the  quality  or  fineness  of  this  composition  1 

Ans.  6  oz.  I3pwt.  8  gr.  fine. 

8.  Suppose  5  Ib.  of  gold  of  22  carats  fine,  2  Ib.  of  21 
carats  fine,  and  1  Ib.  of  alloy  be  melted  together;  what  is 
the  quality  or  fineness  of  this  mass  ? 

Ans.  19  carats  fine. 


ALLIGATION    ALTERNATE, 

IS  the  method  of  finding  what   quantity  of  each   of  the 
ingredients  whose  rates  are  given,  will  compose  a  mixture 
of  a  given  rate  ;  so  that  it  is  the  reverse  of  Alligation  Mg^ 
dial,  and  may  be  proved  by  it. 

CASE  I. 

When  the  mean  rate  of  the  whole  mixture,  and  the  rates 
of  all  the  ingredients  are  given,  without  any  limited  quan- 
tity. 

RULE. 

1.  Place  the  several  rates,  or  prices  of  the  simples,  be- 
ing reduced  to  one  denomination,  in  a  column  under  each 
other,  and  the  mean  price  in  the  like  name,  at  the  left  hand. 

2.  Connect,  or  link  the  price  of  each  simple  or  ingredi- 
ent, which  is  less  than  that  of  the  mean  rate,  with  one  or 
any  number  of  those,   which  are   greater  than  the  mean 
rate,  and  each  greater  rate,  or  price,  with  one,  or  any  num- 
ber of  the  less. 

3-.  Place  the  difference,  between  the  mean  price  (or  mix- 
ture rate)  and  that  of  each  of  the  simples,  opposite  to  thql 
rflfp*  with  which  thev  are  rovme*»t«rl. 


ALLIGATION  ALTERNATE.  179 

4.  TJien,  it*  only  one  difference  stands  against  any  rate, 
it  will  be  the  quantity  belonging  to  that  rate,  but  if  there  be 
several,  their  sum  will  be  the  quantity. 

EXAMPLES. 

1.  A  merchant  has  spices,  some  at  9d.  per  Ib.  some  at  Is. 
some  at  2s.  and  some  at  2s.  6d.  per  Ib.  how  much  of  each 
sort  must  he  mix,  that  he  may  seil  the  mixture  at  Is.  8"d. 
per  pound  1 

Ib.      d.  cL  Ib. 

10at9^  f    9. 

4    12  I  Gives  the  d.  \  12dL^     10 

8    W[ Answer}  or      20]243    1     11 

11    303  [30—         83  T 

2.  A  grocer  would  mix  the  following  qualities  of  sugar ; 
viz.  at  10  cents,  13  cents,  and  16  cents  per  Ib. ;  what  quan- 
tity of  each  sort  must  be  taken  to  make  a  mixture  worth 
12  cents  per  pound  1 

Arts.  5  Ib.  at  10  cts.  2  Ib.  at  13  cts.  and  2  Ib.  at  10  cts.  per  Ib. 

3.  A  grocer  has  two  sorts  of  tea,  viz.  at  9s.  and  at  15s. 
jr  Ib.  how  must  he  mix  them  so  as  to  afford  the  composi- 

for  12s.  per  Ib.  1 

-4ns.   He  must  mix  an  equal  quantity  of  each  sort. 

4.  A  goldsmith  would  mix  gold  of  17  carats  fine,  with 
some  of  19,  21,  and  24  carats  fine,  so  that  the  compound 
ntay  be  22  carats  fine  ;  what  quantity  of  each  must  he  take? 

Ans.%  of  each  of  the  first  three  sorts,  and  9  of  the  last. 

5.  It  is  required  to  mix  several  sorts  of  rum,  viz.  at  5s. 
7s.  and  9e.  per  gallon,  with  water  at  Q  per  gallon,  toge- 
ther, so  that  the  mixture  may  be  worth  6s.  per  gallon ;  how 
much  of  each  sort  must  the  mixture  consist  of? 

J  Ant.  1  gal.  of  rum  at  5s.,  1  do.  at  7s.,  6  do.  at  9s.  and  3  gals, 
wafer.  Or,  3  gals,  rum  at  5s.,  6  do.  at  7s.,  I  do.  at  9s.  and 

1  gal.  water.  • 

1  0.  A  grocer  hath  several  sorts  of  sugar,  viz.  one  sort  at  12 
|cts.  per  Ib.  another  at  11  cts.  a  third  at  9  cts.  and  a  fourth 
at  8  cts.  per  Ib. ;  I  demand  how  much  of  each  sort  he  must 
ijmix  together,  that  the  whole  quantity  may  be  afforded  at 
IjlO  cerfts  per  pound  ? 


J80  ALTERNATION   PARTIAL. 

lb.     cts,  Ib.       cts.  lb.  ct». 

"     at  12  Cl  at  12  f  3  at  12 


£2  at    8  [l  at     8  3at    8 

4£/*  ^iws.  3  /&.  of  each  sort.* 

CASE  II. 
ALTERNATION  PARTIAL, 

Or,  when  one  of  the  ingredients  is  limited  to  a  certain 
quantity,  thence  to  find  the  several  quantities  of  the  rest,  hit 
proportion  to  the  quantity  given. 

RULE. 

Take  the  differences  between  each  price,  and  the  mean 
rate,  and  place  them  alternately  as  in  CASE  I.  Then,  as  the 
difference  standing  against  that  simple  whose  quantity  is] 
given,  is  to  that   quantity  :    so  is  each  of  the  other  differ-  ( 
ences,  severally,  to  the  several  quantities  required. 

EXAMPLES. 

1.  A  farmer  would  mix  10  bushels  of  wheat,  at  70  cents 
per  bushel,  with  rye  at  48  cts.  corn  at  36  cts.  and  barley  at 
30  cts.  per  bushel,  so  that  a  bushel  of  the  composition  may 
be  sold  for  38  cts*;  what  quantity  of  each  must  be  taken  ? 

{70  —  ^  8    stands  against  the  given  quan- 
3(0    J10 
30—'  32 

(    2  :     21  bushels  of  rye. 
As  8  :  10  :  :  \  10  :  12£  bushels  of  corn. 
(  32  :  40"  bushels  of  barley. 


*  These  four  answers  arise  From  as  marir  various  ways  of  linking  the 
rates  of  the  ingredients  together. 

Questions  in  this  rule  adfmitof  an  infinite  variety  of  answers :  for  after  the 
quantities  are  found  from  different  methods  of  linking; ;  any  other  numbers  in 
the  same  proportion  between  themselves,  as  the  numbers  which  compose  thn 
answer,  will  likewise  satisfy  the  conditions  of  the  question. 


ALTERNATION  PARTIAL.  181 

2.  How  much  water  must  be  mixed  with  100  gallons  of 
rum,  worth  7s.  6d.  per  gallon,  to  reduce  it  to  6s.  3d.  per 
gallon  1  Ans.  20  gallons. 

3.  A  farmer  would   mix  20  bushels  of  rye,  at  65  cents 
per  bushel,  with  barley  at  51  cts.  and  oats  at  30  cents  per 
bushel ;  how  much  barley  and  oats  must  be  mixed  with  the 
20  bushels  of  rye,  that  the  provender  may  be  worth  41  cts. 
per  bushel  1 

Ans.  20  bushels  of  'barley,  and  61T?r  bushels  of  oats. 

4.  With  95  gallons  of  rum  at  8s.  per  gallon,  I  mixed  other 
rum  at  6s.  8d.  per  gallon,  and  some  water;  then  I  found  it 
stood  me  in  6s.  4d.  per  gallon  ;   I  demand  how  much  rum 
and  how  much  water  I  took  1 

Ans.  95  gals,  rum  at  6s.  Sd.  and  30  gals,  water. 

CASE  III. 

When  the  whole  composition  is  limited  to  a  given  quantity. 
RULE. 

Place  the  difference  between  the  mean  rate,  and  the  se- 
veral prices  alternately,  as  in  CASE  I.  ;  then,  As  the  sum  of 
the  quantities,  or  difference  thus  determined,  is  to  the  given 
quantity,  or  whole  composition  :  so  is  the  difference  of  each 
rate,  to  the  required  quantity  of  each  rate, 

EXAMPLES. 

1.  A  grocer  had  four  sorts  of  tea,  at  Is.  3s.  6s.  and  10s, 
per  Ib.  the  worst  would  not  sell,  and  the  best  were  too  dear; 
he  therefore  mixed  120  Ib.  and  so  much  of  each  sort,  as  to 
sell  it  at  4s.  per  Ib. ;  how  much  of  each  sort  did  he  take  ? 

i — .  6  re  :  60  at  n 

3^  I  2    Ib.   Ib.    )  2  :  20  —  3  I  ^  lh 
6j   1  As  12  :  120  :  :  ]  1  :  10—  6  \  p 
10  -^  3  ^3:30  — 10  J 

Sum,  12  120 


182  ARITHMETICAL  PROGRESSION. 

2.  How  much  water  at  0  per  gallon,  must  be  mixed  with 
wiive  at  90  cents  per  gallon,  so  as  to  fill  a  vessel  of  100  gal- 
lons, which  may  be  afforded  at  60  cents  per  gallon  1 

Am.  33i gals,  water,  and  66|  gals.  wine. 

3.  A  grocer  having  sugars  at  8  cts.  16  cts.  and  24  cts. 
per  pound,  would  make  a  composition  of  240  Ib.  worth  20 
cts.  per  Ib.  without  gain  or  loss  ;  what  quantity  of  each  must 
be  taken  ? 

Ans.  40  Ib.  at  8  cts.  40  Ib.  at  16  cts.  and  160  Ib.  at  24  cts. 

4.  A  goldsmith  had  two  sorts  of  silver  bullion,  one  of 
10  oz.  and  the  other  of  5  oz.  fine,  and  has  a  mind  to  mix 
a  pound  of  it  so  that  it  shall  be  8  oz.  fine  ;    how  much  of 
each  sort  must  he  take  ? 

Ans.  4J  of  5  oz.fme,  and  7-J-  of  10  oz.  fine. 

5.  Brandy  at  J3s.  6d.  and  5s.  9d.  per  gallon,  is  to  be  mixed, 
so  that  a  hhd.  of  63  gallons  may  be  sold  for  12Z.  12s. ;  how 
many  gallons  must  be  taken  of  each  ? 

Ans.  14  gals,  at  5s.  9d.  and  49  gals,  at  3s.  6d. 


ARITHMETICAL  PROGRESSION. 

ANY  rank  of  numbers  more  than  two,  increasing  by 
common  excess,  or  decreasing  by  common  difference,  is 
said  to  be  in  Arithmetical  Progression. 

g    (  2,4,6,8,  «&e.  is  an  ascending  arithmetical  series  : 
\  8,6,4,2,  &c.  is  a  descending  arithmetical  series  : 

The  numbers  which  form  the  series,  are  called  the  terms 
of  the  progression  ;  the  first  and  last  terms  of  which  are 
called  the  extremes.* 

PROBLEM  I. 

The  first  term,  the  last  term,  and  the  number  of  terms 
being  given,  to  find  the  sum  of  all  the  terms. 

*  A  series  in  progression  includes  five  parts,  viz.  the  first  term,  last  term, 
number  of  terms,  common  difference,  and  sum  of  the  series. 

By  having  any  three  of  these  parts  given,  the  other  two  may  be  found,  • 
which  admits  of  a  variety  of  Problems  ;  but  most  of  them  are  best  under- 
stood by  an  algebraic  process,  and  are  here  omitted. 


ARITHMETICAL  PROGRESSION.  183 

RULE. — Multiply  the  sum  of  the  extremes  by  the  number  of 
terms,  and  half  the  product  will  be  the  answer. 

EXAMPLES. 

1.  The  first  term  of  an  arithmetical  series  is  3,  the  last 
term  23,  and  the  number  of  terms  1 1 ;  required  the  sum  of 
the  series. 

234-3—26  sum  of  the  extremes. 
Then  26  x  11-^2=143  the  Answer. 

2.  How  many  strokes  does  the  hammer  of  a  clock  strike 
in  12  hours.  Ans.  78. 

3.  A  merchant  sold    100  yards  of  cloth,  viz.    the  first 
yard  for  1  ct.  the  second  for  2   cts.  the  third  for  3  cts.  &c. 
I  demand  what  the  cloth  came  to  at  that  rate  ? 

Ans.  $5Q£., 

4.  A    man    bought  19  yards  of    linen   in  arithmetical 
progression,  for  the  first  yard  he  gave  Is.  and   for  the  last 
yd.  I/.   17s.   what   did  the   whole   come    to? 

Ans.  £18  Is. 

5.  A  draper  sold  100  yards   of  broadcloth,  at  5  cts.  for 
the  first  yard,  10  cts.  for  the  second,  15  for  the  third,  &c. 
increasing  5  cents    for   every  yard;    what  did  the   whole 
amount  to,  and  what  did  it  average  per  yard  1 

Ans.  Amount  $252^,  and  the  average  price  is  $2,  52  cts* 
5  mitts  per  yard. 

6.  Suppose  144  oranges  were  laid  2  yards  distant  from 
each  other,  in  a  right  line,  and  a  basket  placed  two  yards 
from  the  first  orange,  what  length   of  ground  will   that  boy 
travel  over,  who  gathers  them  up  singly,  returning  with 
them  one  by  one  to  the  basket  1 

Ans.  23  miles,  5  furlongs,  180  yds. 

PROBLEM  II. 

The  first  term, the  last  term,  and  the  number  of  terms  given, 
to  find  the  common  difference. 

RULE. — Divide  the  difference  of  the  extremes  by  the  number 
of  terms  less  1.  and  the  quotient  will  be  the  common  difference. 


184  ARITHMETICAL  PROGRESSION. 

EXAMPLES. 

1.  The  extremes  are  3  and  29,  and  the  number  of  terms 
14,  what  is  the  common  difference  1 

29)  T7  * 

_g  >  Extremes. 

Number  of  terms  less  1  —  1 3)26(2  Ans. 

2.  A  man  had  9  sons,  whose  several  ages  differed  alike, 
the  youngest  was  three  years  old,  and  the  oldest  35  ;    what 
was  the  common  difference  of  their  ages  ? 

Ans.  4  years. 

3.  A  man  is  to  travel  from   New-London    to   a  certain 
place  in  9  days,  and  to  go  but  3  miles  the  first  day,  increa- 
sing every  day  by  an  equal  excess,  so   that  the  last  day's 
journey   may  be  43  miles :    Required  the    daily  increase, 
and  the  length  of  the  whole  journey  ? 

Arts.  The  daily  increase  is  5,  and  the  whole  journey  207 
miles. 

4.  A  debt  is  to  be  discharged  at   16   different  payments 
(in  arithmetical  progression,)  the  first  payment  is  to  be  147. 
the  last  100?. ;  What  is  the   common   difference,  and   the 
sum  of  the  whole  debt  ? 

Ans.  51.  14s.  Sd.  common  difference,  and  912/.   the  whole 
debt. 

PROBLEM  III. 

Given  the  first  term,  last  term,  and  common  difference,  to 
find  the  number  of  terms. 

RULE. — Divide  the  difference  of  the  extremes  by  the  common 
difference,  and  the  quotient  increased  by  1  is  the  number  of  terms. 

EXAMPLES. 

1.  If  the  extremes  be  3  and  45,  and  the  common  differ- 
ence 2  ;  what  is  the  number  of  terms  ?  Ans.  22. 

2.  A  man   going  a  journey,  travelled  the  first  day  five 
miles,  the  last  day  45  miles,  and  each  day   increased  his 
journey  by  4  miles  ;  how  many   days  did   he  travel,  and 
how  far  ? 

Arts.  11  days,  and  the  whole  distance  travelled  275  mil* $. 


UEOMETRIOAL    PROGRESSION".  185 

GEOMETRICAL  PROGRESSION, 

IS  when  any  rank  or  series  of  numbers  increase  by  one 
^common  multiplier,  or  decrease  by  one  common  divisor; 
as,  1,2,  4,  8,  16,  &-c.  increase  by  the  multiplier  2  ;  and  27, 
9,  3,  1,  decrease  by  the  divisor  3. 

PROBLEM  I. 

The  first  term,  the  last  term  (or  the  extremes)  and  the  ra- 
tio given,  to  find  the  sum  of  the  series. 

RULE. 

Multiply  the  last  term  by  the  ratio,  and  from  the  pro- 
duct subtract  the  first  term  ;  then  divide  the  remainder  by 
the  ratio,  less  by  1,  and  the  quotient  will  be  the  sum  of  all 
the  terms, 

EXAMPLES. 

1.  If  the  series  be  2,  6,  18,  54, 162,  486,  1458,  and  the 
ratio  3,  what  is  its  sum  total  1 

3x1458—2 

— =2186  the  Answer. 

3—1 

2.  The  extremes  of  a  geometrical  series  are  1  and  65536, 
and  the  ratio  4  ;  what  is  the  sum  of  the  series  1 

Ans.  87381. 
PROBLEM  II. 

Given  the  first  term,  and  the  ratio,  to  find  any  other  term 

assigned.* 

CASE  I. 
When  the  first  term  of  the  series  and  the  ratio  are  equal. f 


common  ainerencc  13  i. 

t  When  the  first  term  of  the  series  and  the  ratio  are  equal,  the  indices 
must  begin  with  the  unit,  and  in  this  .case,  the  product  of  any  two  terms  ia 
equal  to  that  term,  signified  by  the  sum  of  their  indices  : 


186  GEOMETRICAL  PROGRESSION. 

1.  Write  down  a  few  of  the  leading  terms  of  the  series, 
and  place  their  indices  over  them,  beginning  the  indices 
with  a  unit  or  1. 

2.  Add  together  such  indices,  whose  sum  shall  make  up 
the  entire  index  to  the  sum  required. 

3.  Multiply  the  terms  of  the  geometrical  series  belonging 
to  those  indices  together,  and  the  product  will  be  the  term 
sought. 

EXAMPLES. 

1.  If  the  first  be  2,  and   the   ratio  2;  what  is  the  13th 
term  ? 

1,2,3,    4,    5,  indices.  Then  5  +  5+3=13. 

2,  4,  8,  16,  32,  leading  terms.      32x32x8=8192  Ans. 

2.  A  draper  sold  20  yards  of  superfine  cloth,  the  first 
yard  for  3d.,  the  second  for  9d.,  the  third  for  27d.,  &c.  in 
triple  proportion  geometrical ;  what  did  the  cloth  come  to 
at  that  ratel 

The  ,'M)th,  or  last :term,is  3^66784401^. 
Then  3+3486784401—3 

=5230176600^.  the  sum  of  all 

3—1 
the  terms  (by  Prob.  I.)  equal  to  £21792402,  10s. 

3.  A  rich  miser  thought  20  guineas  a  price  too  much  for 
12  fine  horses,  but  agreed  to  give  4  cts.  for  the  first,  16  cts. 
for  the  second,  and  64  cents  for  the  third  horse,  and  so 
on  in  quadruple  or  fourfold  proportion  to  the  last :  what 
did  they  come  to  at  that  rate,  and  how  much  did  they  cost 
per  head  one  with  another  1 

Ans.    The  12  horses  came  to  $223696,  20  cts.,  and  the 
average  price  was  $18641,  35  cts.  per  head. 


123     4     5,  &c.  indices  or  arithmetical  series. 
2  4  8  16  32,  <fec.  geometrical  series. 
3+2  =     5  r=    the  index  of  the  fifth  term,  and 
>w»     4x8  =  32  ~    the  fifth  term. 


GEOMETRICAL  PROGRESSION.  187 

CASE  II. 

When  the  first  term  of  the  series  and  the  ratio  are  diffe- 
rent, that  is,  when  the  first  term  is  either  greater  or  less 
than  the  ratio.* 

1.  Write  down  a  few  of  the  leading  terms  of  the  series, 
and  begin  the  indices  with  a  cipher :     Thus,  0,  1 ,  2,  3,  &c 

2.  Add  together  the  most  convenient  indices  to  make  an 
ndex.less  by  1  than  the  number  expressing  the  place  of  the 
erms  sought. 

3.  Multiply  the  terms  of  the  geometrical  series  together 
>elongingto  those  indices,  and  make  the  product  a  dividend. 

4.  Raise  the  first  term  to  a   power  whose  index  is  one 
ess  than  the  number  of  the  terms  multiplied,  and  make  the 
•esult  a  divisor. 

5.  Divide,  and  the  quotient  is  the  term  sought. 

EXAMPLES. 

4.  If  the  first  of  a  geometrical  series  be  4,  and  the  ratio 
2,  what  is  the  7th  term  1 
0,     I,       2,         3,  Indices, 
4,  12,   '36,     108,  leading  terms. 

3 +  2 -{-1=6,  the  index  of  the  7th  term. 
108x36x12—46656 

— —2916  the  7th  term  required. 
16 

Here  the  number  of  terms  multiplied  are  three;  there- 
fore the  first  term  raised  to  a  power  less  than  three,  is  the 
2d  power  or  square  of  4=16  the  divisor. 


*  When  the  first  term  of  the  series  and  the  ratio  are  different,  the  indices 
nust  begin  with  a  cipher,  and  the  sum  of  the  indices  made  choice  of  must 
36  one  less  than  the  number  of  terms  given  in  the  question  :  because  1  in 
Jie  indices  stands  over  the  second  term,  and  2  in  the  indices  over  the  third 
,erm,  &c.  and  in  this  case,  the  product  of  any  two  terms,  divided  by  the  first 
s  equal  to  that  term  beyond  the  first,  signified  by  the  sum  of  their  indices. 
Th  (  0,  1,  2,  3,fc  4,  &c.  Indices. 

\  1,  3,  9,  27,  81,  &c.  Geometrical  series. 
Here  4+3=7  the  index  of  the  8th  term. 
81x27=2187  the  8th  term,  or  the  7th  beyond  the  1st. 


188  POSITION. 

5.  A  Goldsmith  sold  1  Ib.  of  gold,  at  2  cts.  for  the  first 
ounce,  8  cents  for  the  second,  32  cents  for  the  third,  &c.  in 
a  quadruple  proportion  geometrically :  what  did  the  whole 
come  to?  Ans.  $111848,  10  cts. 

6.  What  debt  can  be  discharged  in  a  year,  by  paying  1 
farthing  the  first  month,  10  farthings,  or  (2^d)  the  second, 
and  so  on,  each  month  in  a  tenfold  proportion  ? 

Ans.  £115740740  14s.  9d.  3  qrs. 

7.  A  thrasher  worked  20  days  for  a  farmer,  and  received 
for  the  first  days  work  four  barley-corns,  for  the  second  12 
barley  corns,  for  the  third  36  barley  corns,  and  so   on,  in 
triple    proportion  geometrically.     I   demand  v/hat  the  20 
day's  labour  came  to  supposing  a  pint  of  barley  to  contain 
7680  corns,  and  the  whole  quantity  to  be  sold  at  2s.  6d.  per 
bushel?  Ans.  £1773  7s.  6d.  rejecting  remainders. 

8.  A  man  bought  a   horse,   and   by  agreement,    was  to 
give  a  farthing  for  the  first  nail,   two   for  the   second,  four 
for  the  third,  &c.  There  were  four  shoes,  and  eight  nails  in 
each  shoe  ;  what  did  the  horse  come  to  at  ihat  rate  ? 

Ans.  £4473924  5s.  3jd. 

9.  Suppose  a  certain  body,  put  in  motion,  should  move 
the  length  of  1  barley-corn  the    first   second  of  time,  one 
inch  the  second,  and  thiee  inches  the  third  second  of  time, 
and  so  continue  to  increase  its  motion   in  triple  proportion 
geometrical ;   how  many  yards  would  the  said  body  move 
in  the  term  of  half  a  minute. 

Ans.  953199885623  yds.  1  ft.  I  in.   Ib.  which  is  no   less 
than  five  hundred  and  forty- one  millions  of  miles. 


POSITION. 

POSITION  is  a  rule  which,  by  false  or  supposed  num- 
bers, taken  at  pleasure,  discovers  the  true  ones  required. — 
It  is  divided  into  two  parts,  Single  or  Double. 

SINGLE  POSITION 

IS  when  one  number  is  required,  the  properties  of  which 
are  given  in  the  question. 


POSITION,  189 

RULE. — 1.  Take  any  number  and  perform  the  same  operation 
with  it,  as  is  described  to  be  performed  in  the  question. 

2.  Then  say;  as  the  result  of  the  operation  :  is  to  the  given 
sum  in  the  question  :  :  so  is  the  supposed  number  :  to  the  true 
one  required. 

The  method  of  proof  is  by  substituting  the  answer  in  the  ques 
I  tion. 

EXAMPLES. 

1.  A  schoolmaster  being  asked  how  many  scholars  he 
had,  said,  If  I  had  as  many  more  as  I  now  have,  half  as 
many,  one-third,  and  one  fourth  as  many,  I  should  then 
have  148  ;  How  many  scholars  had  he  ? 

Suppose  he  had  12  As  37  :  148  :  :  12  :  48  An*. 

as  many  =  12  48 

*   as  many  =6  24 

^  as  many  =4  16 

•J   as  many          3  12 

Result,  37  Proof,  148 

2.  What  number  is  that  which  being  increased  by  |,  £, 
and  I  of  itself,  the  sum  will  be  125 1  Ans.  60. 

3.  Divide  93  dollars  between  A,  B   and  C,  so  that  B's 
share  may  be  half  as  much  as  A's,  and  C's  share  three  times 
as  much  as  B's. 

Ans.  A's  share  $31,  B's  $15£,  and  C's  $46|. 

4.  A,  B  and  C,  joined  their  stock  and  gained  360  dols. 
of  which  A  took  up  a  certain  sum,  B  took  3±  times  as  much 
as  A,  and  C  took  up  as  much  as  A  and  B  both  ;  what  share 
of  the  gain  had  each  ? 

Ans.  A  $40,  B  $140,  and  C  $180. 

5.  Delivered  to  a  banker  a  certain  sum  of  money,  to  re- 
ceive interest  for  the  same  at  61.  per  cent,  per  annum,  sim- 
ple interest,  and  at  the  end  of  twelve  years  received  7317. 
principal  and  interest  together ;  what  v/as  the  sum  deliver- 
ed to  him  at  first?  Ans.  £425. 

6.  A  vessel  has  3  cocks,  A,  B  and  C  ;  A  can  fill  it  in  1 
hour,  B  in  2  hours,  and  C  in  4  hours  ;  in  what  time  will 
they  all  fill  it  together?  Ans.  34w??w.  17-V.w. 


190  DOUBLE  POSITION. 

DOUBLE  POSITION, 

. 

TEACHES  to  resolve  questions  by  making  two  swppo 
sitions  of  false  numbers.* 

RULE. 

1.  Take  any  two  convenient  numbers,  and  proceed  with 
each  according  to  the  conditions  of  the  question. 

2.  Find  how  much  the  results  are  different  from  the  re- 
sults in  the  question. 

3.  Multiply  the  first  position  by  the  last  error,  and  the  last 
position  by  the  first  error. 

4.  If  the  errors  are  alike,  divide  the  difference  of  the  pro-  ? 
ducts  by  the  difference  of  the  errors,  and  the  quotient  will 
be  the  answer. 

5.  If  the  errors  are  unlike,  divide  the  sum  of  the  pro-    j 
ducts  by  the  sum  of  the  errors,  and  the  quotient  will  be   i 
the  answer. 

NOTE. — The  errors  are  said  to  be  alike  when  they  are  I 
both  too  great,  or  both  too  small ;  and  unlike,  when  one  I 
is  too  great,  and  the  other  too  small. 

EXAMPLES. 

1.  A  purse  of  100  dollars  is  to  be  divided  among  4  men,  ? 
A,  B,  C  and  D,  so  that  B  may  have  four  dollars  more  than 
A,  and  C  8  dollars  more  than  B,  and  D  twice  as  many  as 
C ;  what  is  each  one's  share  of  the  money  1 

1st.  Suppose  A     6  2d.  Suppose    A     8 

B  10  B  12 

C  18  C  20 

D  36  D  40 

70  80 

100  100 

1st  error,     30  2d  error,     20 


*  Those  questions  in  which  the  results  are  not  proportional  to  their  posi-  ¥ 

tions,  belong  to  this  rule  ;  such  as  those  in  which  the  number  sought  is  in-  I 

creased  or  diminished  by  some  given  number,  which  is  no  known  part  of  the  • 
number  required. 


DOUBLE  POSITION. 


191 


The  errors  being  alike,  are  both  too  small,  therefore, 
Pos.  Err. 
6        30 


X 


8        20  .  Proof  100 

1240       120 
120 

10)120(12  A's  part. 

2.  A,  B,  and  C,  built  a  house  which  cost  500  dollars,  of 
vhich  A  paid  a  certain  sum  ;  B  paid  10  dollars  more  than 
L,  and  C  paid  as  much  as  A  and  B  both  ;  how  much  did 
ach  man  pay  1 

Ans.  A  paid  $120,  B  $130,  and  C$250. 

3.  A  man  bequeathed  100Z.  to  three  of  his  friends,  after 
this  manner  ;  the  first  must   have  a  certain  portion,  the  se- 
cond must  have  twice  as  much  as  the  first,  wanting  SL  and 
the  third  must  have  three  times  as  much  as  the  first,  want- 
ing 15Z. ;  I  demand  how  much  each  man  must  have  ? 

Ans.  The  first  £20  10s.  second  £33,  third  £46  10s. 

4.  A  labourer  was  hired  for  6g  days  upon  this  condition  ; 
that  for  every  day  he  wrought  he  should  receive  4s.  and  for 
every  day  he  was  idle  should  forfeit  2s. ;  at  the  expiration 
of  the  time  he  received  71.  10s. ;     how  many   days   did  he 
work,  and  how  many  was  he  idle  1 

Ans.  He  wrought  45  days,  and  was  idle  15  days. 

5.  "What  number  is  that  which  being  increased  by  its  \ , 
its  J,  and  18  more,  will  be  doubled  1  Ans.  72. 

6.  A  man  gave  to  his  three*  sons  all  his  estate  in  money, 
viz.  to  F  half,  wanting  50?.,  to  G  one-third,  and  to  H  the 
rest,  which  was  10Z.  less  than  the  share  of  G  ;    I  demand 
the  sum  given,  and  each  man's  part  ? 

Ans.  the  sum  given  was  £360,  whereof  F  had  £130,  G 
£120,  and  H  £110. 


192  PERMUTATION  OF  QUANTITIES. 

7.  Two  men,  A  and  B,  lay  out  equal  sums  of  money  in 
trade  ;  A  gains  1267.   and  B  loses  877.  and  A's  money  is 
now  double  to  B's  ;  what  did  each  lay  out  ? 

Ans.  £300. 

8.  A  farmer  having  driven  his  cattle  to  market,  received 
for  them  all  130/.  being  paid  for  every  ox  71.  for  every  cow 
51.  and  for  every. calf  11.    10s.  there  were  twice   as  many 
cows  as  oxen,  and  three  times  as  many  calves   as  cows; 
how  many  were  there  of  each  sort  ? 

'   Ans.  5 oxen,  10  cows,  and  30  Calves. 

9.  A,  B,  and  C,  playing  at  cards,  staked   324   crowns  ; 
but  disputing  about  tricks,  each  man  took  as  many  as   he'< 
could;  A  got  a  certain  number;  B   as   many  as  A  and  15 
more  ;  C  got  a  5th  part  of  both  their  sums  added  together ; 
how  many  did  each  get? 

Ans.  A  got  127|,  B 142J,  C  54. 


PERMUTATION  OF  QUANTITIES, 

IS  the  showing  how  many  different  ways  any  given  num- 
ber of  things  may  be  changed. 

To  find  the,  number  of  Permutations,  or  changes,  that/ 
can  be  made  of  any  given  number  of  things  all  different 
from  each  other. 

RULE.— Multiply  all  the  terms  of  the  natural  series  of  numbers 
from  one  up  to  the  given  number,  continually  together,  and  the  hist 
product  will  be  the  answer  required. 


EXAMPLES. 


1.  How  many  changes  can  be 


made  of  the  first  three  letters  of 

the  alphabet?  Proofj 


a  b  c 
a  c  b 
b  a  c 
b  c  a 
c  b  a 
cab 


.  ^~ 

How  many  changes  may  be  rung  on  9  bells  ? 

Ans.  362880, 


3.  Scvcii  gentlemen  met  at  an  inri;  aiid  Wcte  sp  w«ell 
pleaded  with  their  host,  and  with  eacli  other,  that  they 
agreed  to  tarry  so  long  as  they,  together  with  their  host,*. 
could  sit  every  day  in  a  different  position  at  dinner  ;  how 
Jong  must  they  have  staid  at  said  inn  to  have  fulfilled  their' 
agreement  ?  An$.  HO^-f  years. 

ANNUITIES  OB,  PENSIO3S$i 

COMPUTED  AT 

COMPOUND  INTEREST, 

CASE  t 

To  find  the  auiocrat  of  an  Annuity,  or  iPc'issiqi'i,  iii 
nt  Compound  Interest, 

RUL& 

1.  Make  i  the  first  term  of  a  gedttfetrlcal 

and  the  amount  of  $1  or  £1  for  one  year,  nt  the  give'a  rate' 
pjer  cent,  the  ratio. 

2.  Cf*rry  on  the  scries  irp  to  rig  many  tcritf  s  as  tit}  giyefl 
number  of  years,  and  find  its  sum. 

3.  Multiply  the  snrn  thus  found,  by  the   given  a*mluity», 
and  the  product  will  be  the  amount 


EXAMPLES. 

1.  If*  125  dols.  yearly  rent,  or  annuity  «  he  fo'rbjome  (qr 
unpaid)  4  years  ;  what  will  it  amount  to  at  6  per  cent.  j*e.r 
annum,  compound  interest! 

1  +  1,06+1,12364-1,191016^4,37461(3,  sitm  of  the  s/> 
6s.*  -  -Then,   4,374616  x  125=-$546,827,  t!i«  amount 
sought* 

OR  BY  TABLE  IT. 

Multiply  the  Tabular  number  under  the  rate,  Uud  ojjpb- 
site  to  the  time,  by  the  annuity,  and  the  product  will  bo 
the  amount  sought, 


*  The  sum  of  the  series  thus  found*  is  the  amount  of  ll.  or  1  Hollar  ai\? 
nuity,  for  the  given  time,  which  may  bo  found  in  Table  II.  ready  calcula- 
ted, 

Hignei,  either  the  amount  or  present  wSrtn  of  annuities  miy  PC  readily 
fotmcl  by  tables  for  th»*.  purroso. 

R 


2.  If  a  salary  at  60  dollars  per  annum  tp  he  paid  yearh;, 
be  forborne  twenty  years,  at  6  per  cent,  compound  interest, 
•what  is  tire  amount  ? 

Under  6  pel-  ceril>.  and  opposite  SD-,  in  Table  !£•,  you 
\vill  find, 

Tabular  mi niber -3 6,76559 

60  AnnuTtv. 


13  eft.  5m.+ 

3.  Suppose  an  annuity  of  1007.  be  l&ycarB  in  arrears,  it  is 
required  to.  find  what  is.  now  due,  compound  interest  being 
-allowed  at  5/.  per  cent,  per  annum  1 

'  ^5.  £1.591  145,  0,024^  (by  TaHe  II.) 

4.  \Vhat  will,  a  pennon  of  120?.  per  annum,  payable 
yearly  tampunt.  to  in  3  year*,  at  5L  per  cent,  compound  in- 
ferest  1  Ms.  £378  6^-. 

II.  To  find  tlie  present  w'ortli  of  annuities  at  Compmtnd  In- 
terest. 

tlULE. 

l^vidc  llue  annuity,  &c.  by  that  power  of  tbe  r/itio  sij>> 
nified  by  the  number  of  years,  and  subtract  tbe  quotient 
from  the  annuity  :  This  remainder  being  divided  bj  the  ra- 
tio less  1,  the  quotient  will  be  the  present  value  of  tire  an- 
nuity swght. 


1.  What  ready  nxoney  will  purchase  an  annuity  of  507, 
to  continue  4  years,  at  5?.  per  cent,  compound  interest  ? 


From  50 

Subtract          41,13513' 


Divis.  1,05-1^05)86487 

.  Arts 


QR 

t 

BY  TABLE  III. 

Under  5  per  cent,  and  even  with  4  years., 
We  have  3,54595~present  worth  qf  IL  for  4  years. 
Multiply  by          50=Annuity. 

Ans.  £177,2975Q=present  Worth  of  die  annuity. 
&  What  is  the   present  worth  of  an  annuity  of  60  do'ls-. 
per  annum,  to  continue  £0  years,  at  &  per  cent,  compound 
interest  1  Ans.  $688,  10J tts.  + 

3.  What  is  30Z.  per  annum,  to  continue  7  years]!  worjth  in 
feady  money,  at  6  per  cent,  compound  interest  1 

Ans*  £167  $5.  5d,+ 

III.  To  find  the  present  worth  of  Annuities,  Lease's,  &c,  ta* 
ken  in  REVERSION  at  Compound  Interest. 

1.  Divide  the  annuity  by  that  power  of  the  ratio  denoted 
by  the  time  of  its  continuance. 

2.  Subtract  the  quotient  from  the  annuity :    Divide  the 
remainder  by  the  ratio  less  1,  and  the  quotient  will  be  the 
present  worth  to  commence  immediately. 

3.  Divide  this  quotient  by  that  power  of  tire  ratio  demo- 
ted by  the  time  of  Reversion,  (or  the  time  to  come  be.fore 
the  annuity  commences)  and  the  quotient  will  be  lire  pre- 
sent worth  of  the  annuity  in  .Reversion-, 

EXAMPLES. 

1.  What  ready  mo»ey  will  purchase  .an  annuity  of  50(* 
payable  yearly,  for  4  years ;  but  not  to  commen.ce  till  two 
vears,  at  5  per  cent/? 

4th  power  of  1,05=  1,2 15506) 50,00000(41 ,1&  13 
"Subtract  the  quotient=41, 13513 

Divide  by  1,05— 1 -,05)8,86487 
2d  power  of  1,05-1, 1025)  177,297(  160,8 136^=£  100 
lijTs.  3d.  1  qr.  present  worth  of  the  annuity  in  reversion. 

OR  BY  TABLE  III. 

Find  the  present  value  of  IL  at  the  given  .rate  far  the  sum 
of  the  time  of  continuance,  and  time  in  reversion  added  to- 
gtyher;  from  which  value  subtract  the  present  worth  of  IL 
i'drthe  time  in  reversion,  and  multiply  the  remainder  by  the 
anrmitv  ;  the  nrfydnot  will  he  tire  answer. 


A\Ai;mi^-  OB  r.;: 

Thus  in  Example  1-. 
Time  Of  continuance,  4  years. 
Ditto  of  reversion,        2 

The  sum,  =£  years,  gives  5,0756^ 

fh  A\eve^lQiT4    =S  years,  -    -  1,859410 


Remainder,     3,210282  X  50 

An*.  £160,8141. 

£:  \VJiat  is  the  present  vrorth  of  ?5/.  yearly  rent,  which 
is  not  to  commence  until  10  years  hence,  and  then  to  cftn.- 
iinue  7  years  after  that  time  at  C  per  cent.  ? 

Ans.  £233  15*.  9<L 

4.  What  is  iho  present  worth  of  the  reversion  of  a  lease 
ft?  60  dollars  per  annum,  to  continue  20  years,  but  not  to 
Commence  till  the  endof  B  years,  allowing'  0  per  cent,  to 
the  purchaser  2  Ans.  $431,  78  cts.  2-f^m. 

IV.  To  find  the  present  worth  of  a  Freehold  Estate,  or  an 
Annuity  to  continue  forever,  at  Compound  Interest. 

BULB. 
As  the  rate  per  ccrjt.  is  to  IOO/.  :  sr>  is  the  yearly  rent  to 


J".  Whajt  is  the  worth  of  a  freehold  estate  of  407.  per  fii£ 
nujn,  aHo\vin»-5  pep  cent,  to  the  purchaser  ? 

As^5  :  £100  :  :  £10  :  £800  Ans. 
2.  An  estate  fcrings  in  yearly  1507.  what  would  it  sell  foiY 
allowing  ttio  purchaser  G  per  cent,  for  his  money? 

Ans.  £2500. 
V.  To  find  fhc  present  worth  of  a  Freehold  Estate,  in  Re*- 

versien,  at  Compound  Interest. 

Kur.n.  —  1.  Find  the  present  value  of  the  c&tote  (by  the  foregoing 

.rule)  as  though  it  \vcre  to  be  entered  on  immediately,  and  divide  tlio 

said  vttlae  by  that  power  6f  the  ratio  dsnotod  by  the  lime  of  rcver** 

sion,  und  the  q«6fiefit  \vill  be  the  present  \vorth  of  tlic  estate  in  rc- 

ver^fen. 

EXAMPLES. 

1.  "Suppose  a  freehold  estate  of  407.  per  annum  fo  com*- 
xilenec  two  years  hene"e,.be  put  on  snk-  :  Svhnt  IF  iYs 
allowing  tl/n  pn'rT'Tiris^r  T>/?  rffr  on  ft-.  ? 


ciuES.;riovX5  IAOLI  EXE  uy.  •$;•:,  11J7 


As  &  :  100  ;  :  4Q  :  SOJQ=pre^nt  wqrth  it'  entered  on 
immediately. 

Then,     l,0.5=-l,1025)800,00(7-25,6235S-?-i5/.  12s. 

5^7.=r>resent  worth  of  £800  in  two  years  reversion.  Ans. 

OR  BY  TABLE  III. 

Find  the  present  worth  of  the  annuity,  or  rent,  for  the 
tiiiie  of  reversion,  which  subtract  from  the  value  of  the  im- 
mediate possession,  and  you  will  {lave  the  value  of  the  es- 
tate in  reversion. 

Thus  in  the  foregoing  example, 
k859410i=present  worth  of  l/»  for  2  years. 
40-=  annuity  or  renj. 


74537G40Q=pregent  worth  of  the  aimuily  or  rent,  for 

[the  time  of  reversion. 

From  8(JO,00'00=value  of  immediate  possession. 
Take    74,3764==present  worth  pf  rcn.tf 


2.  Suppose  an  estate  of  90  dollars  per  annum,  to  com* 
mence  10  years  hence,  were  to  be  seld,  gcliowing  thp  pur- 
.cltaser  6  per  cent  ;  what  is  the  worth  \ 

Ans.  $837,  SOcfc.  2w, 

3.  Which  is  the  most  advantageous,  a  term  of  .15  yeaTs, 
iu  an  estate  of  100Z.  per  annum  ;  or  the  reversion  of  such. 
an  estate  forever  after  the  said  15  years,  computing  at  tire 
rate  of  5  per  cent,  per  annum,  compound  interest? 

Ans.  The  first  term  of  15  years  is  better  thtfn  tire  rever* 
s£on  forever  afterwards,tby  £75  18s.  7|*c/. 

A  COLLECTION  OF  QUESTIONS  TO  EXERCISE 
THE  FOREGOING  RULES. 

1.  I  demand  th£  sum  of  1748^  ailed  to  it&el'n 

'Ans.  34'97. 

2.  Whai  i's  the  difference  lietweejii  41  eagles,  and  4U99 
dihies  1  '  Ans.  10  els. 

3.  What  number  is  that  which  btirig  multiplied  by  21, 
the  7»f.tTdfict  will  be  I3BS  ?  An**  65. 


4rKSTio.N>  F<: 

-1.  What  number  is  that  which  being  divided  by  19,  the 
quotient  will  be  72  ?  An*.  1368. 

5.  What  number  is  that  which  being  multiplied  by  15, 
the  product  will  he  J  ?  Ans.  -JF 

(>.  There  are  7  chests  of  drawers,  in  each  of  winch  there 
are  18  drawers,  and  in  each  of  these  there  are  six  divisions, 
in  each  of  which  is  I6/.  6^.  8d.  ;  how  much  money  is  there 
in  the  whole  1  Ans.  £12348.  * 

7.  Bought  3$  pipes  of*  wine  for  4530  dollars  ;  how  mirst 
I  fell  it  a  pipe  to  save  one  for  my  OUT.  use,  and  sell  the  rest 
%  "what  the  whole  cost  ?  Ana.  gl.2f>.  6.0  /;ta 

S.  Just  16  yards  of  German  serge, 
For  DO  dimes  bad  I ; 
ITow  many  yards  of  that  same  cloth 
Will  14  eagles  buy  1         Ans.  246  yds.  3  grs.  2f  nu. 

9.  A  certain  quantity  of  pasture  will  last  963  sheep  7 
•vfceks,  how  many  must  be  turned  out  that  it  will  last  the 
remainder  9  weeks  ?  Ans.  214. 

10.  A  grocer  bought  an  equal  quantity  of  su'gmr,  tea,  and 
coffee,  for  740  dollars  ;   he  gave  10  cents  per  Ib.  for  the  su- 
gar, 60  cts.  per  Ib.  for  the  tea,  and  20  cts.  per  Ib.  for  the 
-collee  ;  required  the  quantity  of  each  ? 

Ans.  822  75. 3  02. 8|  dr. 

11.  Bought  cloth  at  $l£  a  yard,  and  lost  25  per  cent., 
w  was  it  sold  n  yard  1  Ans.  93£  cts. 

12.  The  third  part  of  an  army  was  killed,  the  fourth  pai't 
taken  prisoners,  and  1000  fled  ;  how  many  were  in  this  ar- 
nry,  how  many  killed,  and  bow  many  captives  ? 

Ans.  2400  in  the  army.  800  kitted,  ctnd 

*  600  taken  prisoners. 

13.  Thfcrnas  sold  loO  pine  apples  at  33^  cents  apiece,  and 
received  as  much  money  us  Hnrry  received  for  a  certain 
number  of  water-melons,  which  he  sold  at  25  cents  apiece  ; 
how  much  money  did  each  receive,  and  how  many  melons 
had  Harry?  Ans. Each  rcc'd  $50,  and  Harry  sold  200  melons. 
14.  Said  John  to  Dick,  rny  purse  and  money  are  worth 
9J.  2s. ,  but  the  money  is  twenty-five  times  as  much  as  tile 
purse  ;  I  deifiaml  TiOw  Tmreli  money  was  in  it  ? 

£8  tX<r. 


RU&£tiO&6   rotl  KXKK<  i.-E.  199 

15.  A  young  man  received  210/.  which  was  ~-  of  his  el- 
der  brother's  portion  ;    now  three  times  the  elder  brother's 
portion  was  half  the  father's  estate  ;  what  w*i.s  the  value  of 
the  estate?  Ans.  £1890. 

16.  A  hare  starts  40   yards  before  a  grey-hound,  and  is 
not  perceived  by  him  till  she  has  been  up  40  seconds  ;  she 
scuds  away  at  the  rate  of  ten  miles  an  hour,  and   the  dog, 
on  view,  makes  after  her  at  the  rate  of  18  miles  an  hour  : 
How  long  will  the  course  hold  and  what  space  will  be  run 
over  from  the  spot  where  the  dog  started  ? 

Ans.  60  -o52  sec.  and  530  yds.  space. 

17.  What  number  multiplied  by  57  will  produce  just 
what  134  multiplied  by  71  will  do  1  Ans.  166ff. 

18.  There  are  two  numbers  whose  product   is   1610,  the 
greater  is  given  46  ;  I  demand   the   sum   of  their  squares, 
and  the  cube  of  their  difference? 

Ans.   the  sum    of  their  squares  is    3341.      The  cube,  of 
tJiclr  difference  is  1331. 

19.  Pappose   there  is  a   mast  erected,  so  that  -J-  of  its 
length  stands  in  the  ground,  12  feet  of  it  in  the  water,  and 
|  of  its  length  in  the  air,  or   above  water  ;    I   demand  the 
whole  length  ?  Ans.  SIGfcet. 

20.  What  difference  is  there  between  the  interest  of  5007. 
at  5  per  cent,  for  12  years,   and  the  discount   of  the  same 
sum  at  the  same  rate,  and  for  the  same  time? 

Ans.  £112  10s. 

21.  A  stationer  sold  quills  at  lls.  per  thousand,  by  which 
he  cleared  J  of  the  money,  but  growing  scarce  raised  them 
to  13s.  6d.  per  thousand  :  what   might  he  clear  per  cent. 
by  the  latter  price  ?  "  Ans.  £96  7s.  3T\d. 

22.  Three  persons  purchase  a  West-India  sloop,  towards 
the  payment  *>f  which  A  advanced  f  ,   B  -f  ,  and  C   140/. 
How  much  paid  A   and   B,  and  what  part  of  the   vessel 


Ans.  A  paid  £267T3T,  B  £305T$T,  and  C's  part  of  the 
vessel  was  ££. 

23.  What  is  the  purchase  of  12007.  bank  stock,  at  !(&£ 
percent.  1  Ans.  £1243  16s. 

24.  UotrghfST  pieces  t)f  Nankeens,  each   11^  yard:*,  at 


QUESTIONS 


200 

14s.  4Jd.  a  piece,  which  were  sold  at  18d.  a  yard  ;  required' 
the  prime  cost,  what  it  sold  for,  and  the  gain. 

£.     s.  d. 

c  Prime,  cost,  19     8  1J- 
Ans.  {  Sold  for,     23     5  9 
(  Gain,  3  17  7£ 

25.  Three  partners,  A,  B  and  C,  join  their  stock,  anil 
buy  goods  to  the  amount  of  £1025,5  ;  of  which  A  put  in 
a  certain  sum  ;  B  put  in. ..I  know  not  how  much,  and  C 
the  rest ;  they  gained  at  the  rate  of  24/.  per  cent. :  A's  part 
•of  the  gain  is  -£,  B's  4,  and  C's  the  rest.  Required  each 
man's  particular  stock. 

A's  stock  was  512,7.5 


(  As  stack  was  512,7o 

Aits.   {  &s 205,1 

I  C"s 307,65 


26.  What  is  that  number  which  being  divided  by  £,  the 
•quotient  will  be  21  ?  Ans.  15  j. 

27.  If  to  my  age  there  added  be, 
One-half,  one-third,  and  three  times  three, 
Six  score  and  ten  the  sum  will  be  ; 

What  is  my  age,  pray  show  it  me  1  An.s.  66. 

28.  A   gentleman  divided  his  fortune  among  his   three 
,sons,  giving  A  97.  as  often  as  B  51.  and  to  C  but  3/.  as  often 
as  B  71.  and  yet  C's   dividend   was  2584Z. ;  what   did  the 
whole  estate  amount  to?  Ans.  £19466  2s.  Sd. 

29.  A  gentleman  left  his  son  a  fortune,  |    of  which  he 
spent  in  three  months  ;  %  of  the   remainder  lasted  him  10 
months  longer,  when  he  had  only  2524  dollars  left ;    pray 
what  did  his  father  bequeath  him  ?     Ans.  $5889, 33e^. -f  * 

30.  In  an  orchard  of  fruit  trees,  -J  cf  them  bear  apples, 
£  pears,  £  plums,  40  of  them  peaches,  and    10  cherries  :, 
how  many  trees  does  the  orchard  contain  1        AUK.  600. 

31.  There  is  a  certain  number  which  being  diridet!  by  7, 
the  quotient  resulting  multiplied  by  3,  that  product  divided 
by  5,  from  the  quotient  20  being  subtracted,  and  30  added 
to  the  remainder,  the  half  mi  in  shall  make  65  ;  can  you  teil 

number  t  An*  1400. 


3&  What  part  of  25  is  |  of  a  unit  ?  Ans.  ^V- 

33.  If  A  can  do  a  piece  of  work  alone  in  10  days,  B  in 
^20  days,  C  in  40  days,  and  D  in  80  days;  set  all  four  about 
it  together,  in  whut  time  will  they  finish  it  1  Am.  5£  days. 

34.  A  farmer  being  asked  how  many  bhcep  he  had,  an- 
swered, that  he  had  them  in  live  fields  ;   in  the  first  he  had 
J  of  his  fleck,  in  the  second  £,  in  the  third  {,  in  the  fourth 
T^,  and  in  the  fifth  450  ;  how  many  had  he  \    Ans.  120U. 

35.  A  and  B  together  can  huild  a  hoat  in   18  days,  and 
with  the  assistance  of  C  they  can  do  it  in  1!  days  ;  in  what 
time  would  C  do  it  alone  1  Ans.  281-  days. 

36.  There  are  three  numbers,  23,  25,  and  42;  what  is  the 
difference  between  the  sum  of  the  squares  of  the  first  and 
last,  and  the  cube  of  the  middlemost  I  Ans.  13332. 

37.  Part  1200  acres  of  land  among"  A,  B,  and  C,  so  that 
B  may  have  100  more  than  A,  and  C  64  more  than  B. 

Ans.  A  312,  B  412,  C  476. 

38.  If  3  dozen  pairs  of  gloves  be  equal  in  value  to  2  pieces 
of  Holland,  3  pieces  of  Holland  to  7  yards  of  satin,  6  yards 
of  satin  to  2  pieces  of  Flanders  lace,  and  3  pieces  of  Flan- 
ders lace  to  81  shillings;  how  manv  dozen  pairs  of  gloves 
may  be  bought  for  28s.  ?  Ans.  2  dozen  pairs. 

"39.  A  lets  B  have  a  hogshead  of  sugar  of  18  cwt.,  worth 
5  dollars,  for  7  dollars  the  cwt.  Tf  of  which  he  is  to  pay  in 
cash.  B  hath  paper  worth  2  dollars  per  ream,  which  he 
gives  A  for  the  rest  of  his  sugar,  at  2|  dollars  per  ream  ; 
which  gained  most  by  the  bargain  1  Ans.  A  %  $19  20  cts. 

40.  A  father  left  his  two  sons  (the  one  11  and  the  other 
16  years  old)  10,000  dollars,  to  be  divided  so  that  each  share 
being  put  to  interest  at  5  per  cent,  might  amount  to  equal 
sums  when  they  would  be  respectively  21  years  of  age. 
Required  the  shares?  Ans.  545-1  ~v  and  4545/T  dollars. 

IK  Bought  n  (Vrtain  rwantify  of  broadcloth  for 


'  QU  E S T 1 0$  S  FOil  K  XJiK  c  I  s>  t '. 

5s.  and  if  the  number  of  shillings  which  it  coit  p#r  yard 
were  added  to  the  number  of  yards  bought,  the  sum  would 
he  386 ;  1  demand   the  number  of  yards  bought,  and   at 
what  price  per  yard!         Ans.  365 yds.  at  21s.  per  yard. 
Solved  by  PROBLEM  VI.  page  171* 

42.  Two  partners  Peter  and  John,  bought  goods  to  the 
amount 'of  1000  dollars  ;  in  the  purchase  of  which,  Peter 

paid   more   than  John,  and  John  paid I  know  not   how 

much  :  They  then  sold  their  goods  for^rcady  money,  and 
thereby  gained  at  the  rate  of  200  per  cent,  on  the  prime 
cost :  they  divided  the  gain  between  them  in  proportion  to 
the  purchase  money  that  each  paid  in  buying  the  goods  ; 
and  Peter  says  to  John,  PJy  part  of  the  gain  is  really  a 
handsome  sum  of  money  ;  I  wish  I  had  as  many  such  sums 
as  your  part  contains  dollars, I  should  then  have  $960,000. 
I  demand  each  man's  particular  stock  in  purchasing  tire 
goods.  Ans.  Peter  paid  $600  and  John  paid  £4(M). 

THE  FOLLOWING  QUESTIONS  ARE   EROPOSED  TO  SURVEYORS  I 

1.  Required  to  lay  out  a  lot  of  land  in  form  of  a  long 
square,  containing  3  acres,  2  roods  and  29  rods,  that  shati 
take  just  100  rods  of  wall  to  enclose,  or  fence  it  round  ; 
pray  how  many  rods  in  length,  and  how  many  wide,  must 
said  lot  be?  Ans.  31  rods  in  length,  and  19  in  breadth. 
Solved  by  PROBLEM  VI.  page  171 .  "* 

2L  A  tract  of  land  is  to  he  laid  out  in  form  of  an  equal 
square,  and  to  be  enclosed  with  a  post  and  rail  fence,  5  rails 
high;  so  that  each  rod  offence  shall  contain  10  rails.  How 
large  must  this  noble  square  be  to  contain  just  as  many 
acres  as  there  are  rails  in  the  fence  that  encloses  it,  so  that 
every  rail  shall  fence  an  acre  ? 

Ans.  the  tract  of  land  is  20  miles  square,  and  contains 

256,000  acres. 

Thus,  1  mile=320  rods:  then  320  >  320  -=- 160-640 
acres:  and  320  v 4x10=12,800  rails.  As 640  :  12,800  :  : 
12,800  :  256,000,  rails,  which  will  enclose  £50,000  acres= 
20  nrilcrs  square  >v, 


APPENDIX, 

.    /  CONTAINING 

'SHORT  RULES, 


CASTING  INTEREST  AND  REBATE 

TOGETHER.    WITH    SOME 

USEFUL    RULES, 

'    F.'OR    FINDINCi    TirE    CONTENTS    OF    SUPERFICES,    SOLIT>S,    &C. 


SHORT   RULES, 
FOR  CASTING  INTEREST  AT  SIX  PER  CEN?P. 

I.  To  find  the   interest  of  any  sum  of  shillings   for 
number  of  days  less  than  a  month,  at  6  per  cent. 

RULE. 

1.  Multiply  the  shillings  of  the  principal  by  the  number 
of  days,  and  that  product  by  2,  and  cut  off  three  figures  to 
the  right  hand,  and  all  above  three  figures  will  be  the  interest 
in  ponce. 

2.  Multiply  "the  figures   cut  off  by  4,  still  striking  off 
three  figures  to  the  right  hand,  and  you  will  have  the  far- 
thing's, very  nearly. 

EXAMPLES. 

1.  Required  the  interest  of  51.  8s.  for  25  days. 
£.      s. 

5,8=108x25x2=5,400,  and  400x4=1,600. 

Ans.  5d.  IfiqrS. 

2.  "Wftat  is  tlie  interest  of  217.  3s.  for  29  days  ? 


204  . 

FEDERAL  MONEY. 

II.  To  find  the  interest  of  any  number  of  cents  for  any 
number  of  days  less  than  a  month,  at  G  per  cent. 

RULE. 

Multiply  the  cents  by  the  number  of  days,  divide  the  ptm~ 
duct  by  6,  and  point  off  two  figures  to  the  right,  and  all  tho 
figures  at  the  left  hand  of  the  dash,  will  be  the  interest  ia 
mills,  nearly. 

EXAMPLES. 

Required  the  interest  of  85  dollars,  for  20  days. 
$      cts.  mitls< 

85=8500x20  ^-6^283,33  Am.  283  whicli  i* 

28  cts.  3  mills. 

2.  What  is  the  interest  of  73  dollars  41  cents,  or  7241 
cents,  for  27  days,  at  6  per  cent.  ? 

Ans.  330  miffs,  or  0.3'  cf.*r 


III.  When  the  principal  is  given  in  pounds,  shilling jt,  «fcc. 
New-England  currency,  to  find  the  interest  for  any  num- 
ber of  days,  less  than  a  month,  in  Federal  Money. 

RULE. 

Multiply  the  shillings  in  the  principal  by  the  number  of 
days,  and  divide  the  product  by  36,  the  quotient  will  be  the 
interest  in  mills,  for  the  given  time,  nearly,  omitting 
fractions. 

EXAMPLE. 

Required  the  interest  in  Federal  Money,  of  277.  15s.  for  I 
27  days,  at  6  per  cent. 
£.     s.      s. 
Ans.  27    1 5=555x27-^36=416  milt$.=41  cte.  Gm. 


IV.  When  the  principal  is  given  in  Fecferal  Money,  ahtl 
you  want  the  interest  in  shillings,  pence,  &c.  NW-' 
land  currency*  for  anv  number  tff  d^vs  !c?2  thsri  a  Hv 


APPENDIX.  205 

RULE. 

Multiply  the  principal,  in  cents,  by  the  number  of  days, 
and  point  off  five  figures  to  the  right  hand  of  the  product, 
which  will  give  the  interest  for  the  given  time,  in  shillings 
and  decimals  of  a  shilling,  very  nearly. 

EXAMPLES. 

A  note  for  65  dollars,  31  cents,  has  been  on  interest  25 
days  ;  how  much  is  the  interest  thereof  in  New-England 
currency7?  $  cts.  s.  s.  d.  qrs. 

.  Ans.  65,31=6531  x  25=1,  63275=1  7  2 
REMARKS.  —  In  the  above,  and  likewise  in  the  preceding 
practical  Rules,  (page  115)  the  interest  is  confined  at  6  per 
cent,  which  admits  of  a  variety  of  short  methods  of  cast- 
ing :  and  when  the  rate  of  interest  is  7  per  cent,  as  esta- 
blished in  New-York,  &c.  you  may  first  cast  the  interest  at 
6  per  cent,  and  add  thereto  one  sixth  of  itself,  and  the  sum 
will  be  the  interest  at  7  per  ct.,  which  perhaps,  many  times, 
will  be  found  more  convenient  than  the  general  rule  of  cast- 
ing interest. 

EXAMPLE. 

Required  the  interest  of  751  for  5  months,  at  7  percent. 
.<?. 
7,5  for  1  month. 

5 

-  £.  s.  d. 
37,5=1  17  6  for  5  months  at  6  per  ceut< 


Ans.  £2  3  9  for  ditto  at  7  per  cent. 

A  SHORT   METHOD  FOR  FINDING  THE  REBATE  OF  ANY  GIVEN 
SUM,  FOR  MONTHS  AND  DAYS. 

RULE.  —  Diminish  the  interest  of  the  given  sum  for  the  time  by  its 
own  interest,  and  this  gives  the  Rebate  very  nearly. 

EXAMPLES. 

1.  What  is  the  rebate  of  50  dollars,  for  6  months,  a't  6 
per  cent,  t 


206 


$    Clf. 

The  interest  of  50  dollars  for  G  months,  is  1  50 

And,  the  interest  of  1  dol.  50  cts.  for  6  months,  is          4 


Am.  Rebate,  §1  46 

S.  What  is  the   rebate  of  1507.  for  7  months,  at  5  per 
c.ent.  1  £>.  s.  d. 

Interest  of  1507.  for  7  months,  is  476 

Interest  of  47.  7s.  6d.  for  7  months,  is  2    6£ 


Ans.£<l  4 

By  the  above  Rule,  those  who  use  interest  tables  in  their 
Counting-houses,  have  only  to  deduct  the  interest  of  the  in- 
terest, and  the  remainder  is  the  discount. 


A  concise  Rule  to  reduce  the,  currencies  of  the  different  States, 
where  a  dollar  is  an  even  number  of  shillings,  to  Federal 
Honey. 

R.ULE.  I.  —  Bring  the  given  sum  into  a  decimal  expression  by  in- 
spection, (as  in  Problem  I.  page  <>0)  tiv.-n  divide  the  whole  by  .5  in 
New-England,  and  by  ,4  in  New-York  currency,  and  the  quotient 
will  be  dollars,  cents,  &c. 

EXAMPLES. 

1.  Reduce  547.  8s.  3^d.  New-England  currency,  to  fo 
deral  money. 

,8)5-1,415  decimally  expressed. 

Ans.  $181,38  cts. 

2.  Reduce  7s.  ll|d.  New-England  currency,  to  federal 
money. 

7s.  lljd.=£0,399  then,  ,3),399 


3.  Reduce  5137.  16s.  lOd.  New-York,  &c.  currency,  to 
federal  rnonev. 

,4)513,842  decimal. 


Ans.  $1284,604 


APPENDIX.  £07 

4.  Reduce  19s.  5jd.  New-  York,  &c.  currency,  to  Fede* 
ral  Money.  ,4)0,974  decimal  of  19s.  5Jd." 

$2,431  Ans. 

5.  Reduce    647.    New-England    currency,   to   Federal 
Money.  ,3)64000  decimal  expression. 

$213,331  Ans. 

NOTE.  —  By  the  foregoing  rule  you  may  carry  on  the  de-' 
cimal  to  any  degree  of  exactness  ;  but  in  ordinary  practice, 
the  following  Contraction  fnay  be  useful. 


RULE  II. 

To  the  shillings  contained  in  the  given  sum,  annex  6 
times  the  given  pence,  increasing  the  product  by  2  ;  then 
divide  the  whole  by  the  number  of  shillings  contained  in  a 
dollar,  and  the  quotient  will  be  cents. 

EXAMPLES. 

1.  Reduce  45s.  6d.  New-England  currency,  to  Federal 
Money.  6  x  8-f  2  ==  50  "to  be  annexed. 

6)45,50  or  6)4550    . 

________        __  #  p[$t 

$7,58-1  Ans.     758  cents.—  7,58 

2.  Reduce  2/.    10s.   9d.   New-York,  &c.   currency,  to 
Federal  Money. 

9x84-2=74  to  be  annexed. 
Then  8)5074  Or  thus,  8)50,74 


Ans.       634  cents.=.6  34  $6,34 

N.  B.  When  there  are  no  pence  in  the  .given  sum,  you 
must  annex  two  ciphers  to  the  shillings  ;  then  divide  as  be- 
fore, &c. 

3.  Reduce  3Z.  5s.  New-England  currency,  to  Federal 
Money, 

M.  5&=65s.     Then  6)6500 

Ans.      1083  ceitf*. 


203  APPENDIX. 

SOME  USEFUL  RULES, 

FOR    FINDING    THE    CONTENTS    OF    SUPERFICES    AND    SOLIDS.. 

SECTION  I.— OF  SUPERFICES. 

The  superfices  or  area  of  any  plane  surface,  is  comrw> 
sed  or  made  up  of  squares,  either  greater  or  less,  according 
to  the  different  measures  by  which  the  dimensions  of  the 
figure  are  taken  or  measured: — and  because  12  inches  in 
length  make  1  foot  of  long  measure,  therefore,  12  X  12==  144 
fhe  square  inches  in  a  superficial  foot,  &e. 

ART.  I.  To  find  the  area  of  a  square  having  equal  sides, 

RULE. 

Multiply  the  side  of  the  square  into  itself  and  the  pro* 
duct  will  be  the  area,  or  content. 

EXAMPLES. 

1 .  How  many  square  feet  of  boards  are  contained  in  the 
floor  of  a  room  which  is  20  feet  square  ? 

20     20=400  feet,  the  Answer. 

2.  Suppose  a   square  lot  of  land  measures  26  rods  on 
each  side,  how  many  acres  doth  it  contain  ? 

NOTE.— 160  square  rods  make  an  acre. 

Therefore,  26x26=676  sq.  rods,  and  676-^160=4^ 

36  r.  the  Answer. 
ART.  2.  To  measure  a  parallelogram,  or  long  square. 

RULE. 

Multiply  the  length  by  the  breadth,  arid  the  product  will 
be  the  area,  or  superficial  content. 

EXAMPLES. 

1.  A  certain  garden,  in  form  of  a  long  square,  is  96  feet 
long,  and  54  wide  ;  how  many  square  feet  of  ground  are 
contained  in  it  ?•  Ans.  96  X  54—5184  square  feet. 

2.  A  lot  of  land,  in  form  of  a  long  square,  is  120  rods  in 
length,  and  60  rods  wide  ;  how  many  acres  are  in  it  ? 

120  ^ :' 60=7200  sq.  rods,  then  ^^^  acres.  Ans. 
9.  If  a  board  or  plank  be  21  feet  long,   and    18  inches 
brcrad  ;  how  many  square  feet  are  contained  in  it  ? 

18  mckes*=l,5feet,  then,  21 X  1.5-31,5.    Am, 


Ai'l'liNDIX.  2GU 

Or,  in  measuring  boards,  you  may  multiply  the  length  iu 
Feet  by  the  breadth  in  inches,  arid  divide  by  12,  the  quo- 
tient will  give  the  answer  iu  square  feet,  &c. 

Thus,  in  the  foregoing  example,  21  X  18 -i- 12=3 J  ,5  as 
before. 

4.  If  a  board  be  8  inches  wide,  how  much  in  length  will 
make  a  square  foot  1 

RULE.— Divide  144  by  the  breadth,  thus,      '     8)144 

Ans.     1*3  in. 

5.  If  a  piece  of  land  be  5  rods  wide,  how  many  rods  in 
length  will  make  an  acre? 

RULE. — Divide  160  by  t^c  breadth,  and  the  quotient  will  bo  the 
length  required,  thus, 

5)160 

Ans.     3*2  rods  in  length. 
ART.  3. — To  measure  a  triangle. 

Definition. — A  triangle  is  any  three  cornered  figure  which 
is  bounded  by  three  right  lines.* 

RULE. 

Multiply  the  base  of  the  given  triangle  into  half  its  per- 
pendicular height,  or  half  the  base  into  the  whole  perpen- 
dicular, and  the  product  will  be  the  area. 

EXAMPLES. 

1.  Required  the  area  of  a  triangle  whose  base  or  longest 
side  is  32  inches,  and  the  pcipen^  igki   I -i  i  fches, 

;V2     7— 2ui  swart  inches  the  Answer. 

2.  There  is  a  triangular  ;-jnrn:-red  lot  of  b  :d  whose 
base  or  longest  side  is  5!  be  perpendicular  from  die 
corner  opposite  the  base  measures  44  rods  :  how  many  acres 
doth  it  contain  1 

51,5-22—1133  square  rods,=7  acres,  13  rods. 


*  A  Triangle  may  be  either  right  angled  or* oblique  ;  in  either  case  the 
teacher  can  easily  give  the  scholar  a  right  idea  of  the  base  and  perpcnriicu- 
.far*  By  .marking  it  down  on  the  slate,  paper,  &c~. 


21 0 

TO  MEASURE  A  CIRCLE. 

ART.  4. — The  diameter  of  a  circle  being  given,  to  find 
the  circumference. 

RULE. — As  7  :  is  to  22  :  :  so  is  the  given  diameter  :  to  the  circum- 
ference. Or,  more  exactly,  as  113  :  is  to  355  :  :  £c.  the  diameter  is 
found  inversely. 

NOTE. — The  diameter  is  a  right  line  drawn  across  the 
circle  through  its  centre. 

EXAMPLES. 

1.  What  is  the  circumference  of  a  wheel  whose  diameter 
is  4  feet  1 — as  7  :  22  :  :  4  :   12,57  the  circumference. 

2.  What  is  the  circumference  of  a  circle  whose  diameter 
is  35?— As  7  :   22  :  :  35  :  110  Ans.— and    inversely  as 
22  :  7  :  :   110  :  35,  the  diameter,  &c. 

ART.  5. — To  find  the  area  of  a  Circle. 

RULE. — Multiply  half  the  diameter  by  half  the  circumference,  and 
the  product  is  the  area;  or  if  the  diameter  is  given  without  the  cir- 
cumference, multiply  the  square  of  the  diameter  by  ,7854,  and  the 
product  will  be  the  area. 

EXAMPLES. 

1.  Required  the  area  of  a  circle  whose  diameter  is  12 
inches,  and  circumference  37,7  inches.  ^ 

18,85^=half  the  circumference. 
6— half  the  diameter. 


113,10  area  in  square  inches. 

S.  Required  the  area  of  a  circular  garden  whose  diame- 
ter is  11  rods?  ,7854 
By  the  second  method,  11x11  =^  121 

Ans.  95,0334  rods-. 
SECTION  2.— OF  SOLIDS. 

Solids  are  estimated  by  the  solid  inch,  solid  foot,  &c. 
1728  of  these  inches,  that  is,  12  X  12  X  12  make  1  cubic  or 
fb.ot. 


_  211 

AUT.  6. — To  measure  a  Cube. 

Definition. — A  cube  is  a  solid  of  six  equal  sides,  each  of 
which  is  an  exact  square. 

RULE. — Multiply  the  side  by  itself,  and  that  product  by  the  same 
side,  and  this  last  product  will  be  the  solid  content  of  the  cube. 

EXAMPLES. 

1.  The  side  of  a  cubic  block  being  18  inches,  or  1  foot 
and  6  inches,  how  many  solid  inches  doth  it  contain  ? 

ft.  in  ft. 

1  6=1,5  and  1,5  X  1,5  x  1,5=3,375  solid  feet.    Ans. 
Or,  18  X  18  x  18=5832  solid  inches,  and  f  111=3,375. 

2.  Suppose  a  cellar  to  be  dug  that  shall  contain    12  feet 
every  way,  in  length,  breadth  and  depth ;  how  many  solid 
feet  of  earth  must  be  taken  ouMo  complete  the  same  ? 

12  *  12     12=1728  sold  feet,  the  Ans. 

ART.  7. — To  find  the  content  of  any  regular  solid  of  three 
dimensions,  length,  breadth  and  thickness,  as  a  piece  of 
timber  squared,  whose  length  is  more  than  the  breadth 
and  depth. 

RULE. — Multiply  the  breadth  by  the  depth,  or  thickness,  and  that 
product  by  the  length,  which  gives  the  solid  content. 

EXAMPLES. 

I.  A  square  piece  of  timber,  being  one  foot  6  inches,  or 
18  inches  broad,  9  inches  thick,  and  9  feet  or  108  inches 
long ;  how  many  solid  feet  doth  it  cgntain  ? 

1  ft.     6  ki.=l,5    foot 

9  inches     ---.     ,75  foot. 

Prod.  1,125     9=10,125  solid  feet,  the  Ans. 
in.     in.    in.     solid  in. 
Or  18x9x108=17496     1728=1 0,125 /eef. 

But,  in  measuring  timber,  you  may  multiply  the  breadth 
in  inches,  arid  the  dopth  in  inches,  and  th*it  product  by  the 
length  in  feet,  and  divide  the  last  product  by  144,  which 
will  give  the  solid  content  in  feet,  &c. 


2.  A  piece  of  timber  being  16  inches  broad,  11  inches 
thick,  and  20  feet  long,  to  find  the  content  1 

Breadth  1G  inches. 
Depth     11 

Prod.  176  x  20=3520  then,  3520-  144=&4,4/ee*.  Ans. 

3.  A  piece   of  timber  1.5   inches   broad,  8  inches  thick, 
and  25 feet  long  ;  how  many  solid  feet  doth  it  contain? 

Ans.  20,8-h/ccf. 

ART.  8. — When  the  breadth  and  thickness  of  a  piece  of 
timber  are  given  in  inches,  to  find  how  much  in  length 
will  make  a  solid  foot. 

BJULE. — Divide  1728  by  the  product  of  the  breadth  and  depth,  anil 
the  quotient  will  be  the  length  making  a  solid  foot.  • 

EXAMPLES. 

1.  If  a  piece  of  timber  be  11  inches  broad  and  8  inches 
deep,  how  many  inches  in  length  will  make  a  solid  foot? 

11x8=88)1728(19,6  inches.  Ans. 

2.  If  a  piece  of  timber  be  18  inches  broad  and  14  inches 
deep,  how  many  inches  in  length  will  make  a  solid  foot? 

18  X  14=252  divisor,  then,  252)1728(6,8  inches.    Ans. 

ART.  0. — To  measure  a  Cylinder. 

Definition. — A  Cylinder  is  a  round  body  whose  bases  oca 
-circles,  like  a  round  column  or  suck,  of  \ .-, a) f>er,  of  equal  big- 
ness from  end  to  end. 

RULE, — Multiply  tfee  square  of  t^e  clwriiuter  of  the  end  by  ,7854 
which  gives  the  area  of  the  bab-c  •  i-law  .s-.i-tiply  the  area  of  the  base 
by  the  length,  and  the  product  wii-  be  the  'solid  etm  tent. 

%  EXAMPLE. 

What  is  the  solid  content  of  a  round  stick  of  tiir-^  /  of 
equal  bigness  frouj  end  to  end,  whose  diameter  is  18  inches*, 
length  20 


PPENDiN.  213 

IS  in— 1,5  ft, 
xl,5 


Square  2,25  x  ,7854=1,76715  area  of  the  base. 
i  20  length. 

Ans.      35,34300  solid  content. 
Or,  18  inches. 
18  inches. 

324x,7854=-254,4696  inches,  area  of  the  base- 
20  length  in  feet. 


144)5089,3920(35,343  solid  feet.  Ans. 

ART.  10.  To  find  how  many  solid  feet  a  round  stick  of 
timber,  equally  thick  from  end  to  end,  will  contain  when 
hewn  square. 

RULE. 

Multiply  twice  the  square  of  its  semi-diameter  in  inches 
by  the  length  in  feet,  then  divide  the  product  by  144,  and 
the  quotient  will  be  the  answer. 

EXAMPLE. 

If  the  diameter  of  a  round  stick  of  timber  be  22  inches 
and  its  length  20  feet,  how  many  solid  feet  will  it  contain 
when  hewn  square  ? 

11X11X2  20-M 44=33,6  4-  feet,  the  solidity  when 
hewn  square. 

ART.  11.  To  find  how  many  feet  of  square  edged  boards 
of  a  given  thickness,  can  be  sawn  from  a  log  of  a  given 
diameter. 

RULE. 

Find  the  solid  content  of  the  log,  when  made  square,  by 
the  last  article — Then  say,  As  the  thickness  of  the  board 
including  the  saw  calf  :  is  to  the  solid  feet  :  :  so  is  12  (in- 
ches) to  the  number  of  feet  of  boards. 

KX.  \MPLE. 

How  many  feet  of  square  edged  boards,  \\  inch  thick, 
including  the  saw  calf,  can  be  sawn  from  a  log  20  feet  long 
and  24  inches  diameter  1 

12  X  12     2  X  20  — 144=40  feet,  solid  content. 
As  1  4-  :  40  :  :^M  :  3$4  feet,  thfl  Atfs-. 


ART.  12.  The  length,  breadth  and  depth  of  any  square  box 
being  given,  to  find  how  nmny  bushels  it  will  contain. 

Multiply  the  length  by  the  breadth,  and  that  product  by 
the  depth,  divide  the  last  product  bv  2150,425  the  solid 
inches  in  a  statute  bushel,  and  the  quotient  will  be  the  an- 
swer. 

EXAMPLE. 

There  is  a  square  box,  the  length  of  its  bottom  is  50 
inches,  breadth  of  ditto  40  inches,  and  its  depth  is  60 
inches  ;  how  many  bushels  of  corn  will  it  hold  ? 

50  x  40  x  60-:-2150,425==55,84  r    or  55  bushels  three 
pecks.    Am. 

.AiiT.  13.  The  dimensions  of  the  walls  of  a  brick  building 
being  given,  to  find  how  many  bricks  are  necessary  to 
build  it. 

RULE. 

From  the  whole  circumference  of  the  wall  measured 
round  on  the  outside,  subtract  four  times  its  thickness,  then 
multiply  the  remainder  by  th<^  height,  and  that  product  by 
the  thickness  of  t!u  wall,  gives  the  solid  content  of  the 
whole  wall ;  which  multiplied  by  the  number  of  bricks 
contained  in  a  solid  foot  gives  the  answer. 

EXAMPLE. 

How  many  bricks  8  inches  long,  4  inches  wide,   and  2i 

irichcs  thick,  will  it  take  to  build  a  house  44  feet  long-,  40 

feet  wide,  and  20  feet  high,  and  the  walls  to  he  1  foot  thick  ? 

8x4x2,5=80  solid   inches  in  a  brick,  then  1728^80— 

£1,6  bricks  in  a  solid  foot. 

44  f* 40  f  44  4-40=108  feet,  whole  length  of  wall, 
— 4  times  the  thickness, 

104  remains. 
Multiply  by          20  height. 

3280  solid  feet  in  the  whole  wall-. 
Multiply  by       21,6  bricks  in  a  solid  foot. 

Product.      70848  bricks. 


ART.  14. — To  find  the  tonnage  of  a  ship. 
RULE. — Multiply  the  length   of  the  keel   by  the   breadth   of  the 
beam,  and  that  product  by  the  depth  of  the  hold,  and  divide  the  last 
product  by  95,  and  the  quotient  is  the  tonnage. 

EXAMPLE. 

Suppose  a  ship  72  feet  by  the  keel,  and  24  feet  by  the 
beam,  and  12  feet  deep  ;  what  is  the  tonnage] 

72x24 x  12 -r 95=218,2 + tons.  Ans. 

RULE  II. 

Multiply  the  length  of  the  keel  by  the  breadth   of  the  beam,  anc| 
that  product  by  half  the  breadth  of  the  beam,  and  divide  by  95. 

EXAMPLE. 

A  ship  84  feet  by  the  keel,  28  feet  by  the  beam  ;  what  is 
the  tonnage  ?  84   : 28  *  14-95=350,29  tons.  Ans. 

ART.  15. — From  the  proof  of  any  cable,  to  find  the  strength 
of  another. 

RULE. — The  strength  of  cables,   and  consequently  the  weights  of 
Iheir  anchors,  are  as  the  cube  of  their  peripheries. 
Therefore  ;    As  the  cube  of  the  periphery  of  any  cable, 

Is  to  the  weight  of  its  anchor; 

So  is  the  '-.'libe  of  the  periphery  of  any  other  cable, 

To  the  weight  of  its  anchor. 

EXAMPLES. 

1.  If  a  cable  6  inches  about,  require  an  anchor  of  2£  cwt. 
of  what  weight  must  an  anchor  bt  for  a  12  inch  cable? 

As  6x6     6  :  2^  cwt.  :  :  12     12  v  12  :  IS  cwt.  Ans. 

2.  If  a  12  inch  cable  require  an  anchor  of  18  cwt.  what 
must  the  circumference  of  a  cable  be,  for  an  anchor  of  2| 
cwt.? 

cwt.  cwt.  n  in. 

As  18  :   12     12*12    :  :    2,25  :  216v>2l6r=6   Ans. 
ART.  16. — Having  the  dimensions  of  two  similar  built  ships 
of  a  different  capacity,  with  the  burthen  of  one  of  them-, 
to  find  the  burthen  of  the  other. 


216  APPENDIX. 

RULE. 

The  burthens  of  similar  built  ships  are  to  each  other  j  as 
the  cubes  of  their  like  dimensions. 
EXAMPLE. 

If  a  ship  of  300  tons  burthen  be  75  feet  long  in  the  keel, 
I  demand  the  burthen  of  another  ship,  whose  keel  is  100 
feet  long  1  T.  cwt.  qrs.  Ib. 

As/Tox  75x75: 300  ::  100x100x100:711  2     0    24+ 

DUODEGTMALS, 

OR 

CROSS  MULTIPLICATION, 

IS  a  rule  made  use  of  by  workmen  and  artificers  in  cast- 
ing up  the  contents  of  their  work. 

RULE. 

1.  Under  the  mulplicand  write  the  corresponding  deno-- 
minations  of  the  multiplier. 

2.  Multiply  each  term  into  the  multiplicand,  beginning 
at  the  lowest,  by  the  highest  denomination  in  the  multiplierr 
and  write  the  result  of  each  under  its  respective  term  ;  ob- 
serving to  carry  an  unit  for  every  12,  from  each  lower  de- 
nomination to  its  next  superior. 

8.  In  the  same  manner  multiply  all  the  multiplicand  by 
the  inches,  or  second  denomination,  in  the  multiplier,  and 
set  the  result  of  each  term  one  place  removed  to  the  right  - 
hand  of  those  ir*the  multiplicand. 

4.  Do  the  same  with  the  seconds  in  the  multiplier,  set- 
ling  the  result  of  each  term  two  places  to  the  right  hand  of 
those  in  the  multiplicand,  &c. 


EXAMPLES. 

F.  I.            F.  I.            F.  L 

Multiply  73               75               46 
By            47              39              58 

F.L 

9  7 
9  7 

29  0  "         27  9  9        25  6 
\  o  o               . 

91  10  1 

Product,  33  2  9 


/'.  /. 

Multiply     4     7 
By              5  10 

F.  J. 

3     8 

7     6 

F.   / 

9     7 
3     0 

Product,  26     8  10 

27     6 

32     6    0 

F.  L 

Multiply     3  11 
By              95 

F.  /. 

6     5    . 

7     6 

F.  t 

7  Itf 
8  11 

Product,  36  10  7     48  1  6     69  10  2 


FEET,  INCHES  AND  SECONDS. 

F.  L    " 

Multiply     986  ,     .  . 

By  793 

[tiplier. 

=prod.  by  the  feet  in  the  mul- 
7  46"       =ditto  by  the  inches. 

2516     =ditto  by  the  seconds. 

75     5    3     7     6     Ans. 


F.  L     '  F.  I. 

Multiply     719  567 

By  789  8     9  10 

Product,  55    2     9  3  9  48  11     2   8  10 


How  many  square  feet  in  a  boar4  16  feet  9  inches  long, 
and  2  feet  3  inches  wide  1 

By  Duodecimals.  By  Decimals. 

F.     L  F.     L 

16     9  16     9=16,75  feet. 

23  2     3=2,25 


33     6  8375 

4     2     3  3350 

3350 

W!     8     3 F.    L 

IN*.  37,6875=87     8 


218  APPENDIX. 

TO  MEASURE  LOADS  OF  WOOD. 

RULE. — Multiply  the  length  by  the  breadth,  and  the  product  by  the 
-depth  or  height,  which  will  give  the  content  in  solid  feet ;  of  which  64 
inake  half  a  cord,  and  ]'28  a  cord, 

EXAMrhK:. 

How  many  solid: feet  are  contained  in  a  load  of  wood, 
7  feet  6  inches  long,  4  feet  2  inches  wide,  and  2  feet  3 
inches  high  ? 

7ft.  6  i».= 7,5  /zwd  4  ft.  2  *».  =4,167  0/^  2//.  3  w.= 
2,25  ;  then,  7,5  x  4,167^1^25  x  2,25=70,318125  jo&W 
/<;e#,  /bis. 

But  loads  of  \vood  are  commonly  estimated  by  the  foot, 
allowing  the  load  to  be  8  feet  long,  4  feet  wide,  and  then  2 
feet  high  will  make  half  a  cord,  which  is  called  4  feet  of 
wood ;  but  if  the  breadth  of  the  load  be  less  than  4  feet,  its 
height  must  be  increased  so  as  to  make  half  a  cord,  which 
is  still  called  4  feet  of  wood. 

By  measuring  the  breadth  and  height  of  the  load,  tire 
content  may  be  found  by  the  following 

RuLrE. — Multiply  the  breadth  by  the  height,  and  half  the  product 
will  be  the  content  in  feet  and  inches. 

EXAMPLE. 

Required  the  content  of  a  load  of  wood  which  is  3  feet  9 
inches  wide  and  2  feet  6  inches  high. 
Sy  Duodecimals.   .B?/  Decimals. 

F.  in.  F. 

3    9  3,75 

2    6  2,5 


7     6  1875 

%  10     6  750 


9    4    6  9,375 

F.  in. 


Ans.      -4    8     3  4,6875=4  8j-  or  kalfa  cord  and  8£ 

incjits  over. 

The  foregoing  method  is  concise  and  easy  to  those  who  are  well 
acquainted  with  Duodecimals,  but  the  following  table  will  give  the 
<-onttnt  of  any  load  of  wood,  by  inspection  only,  sufficiently  exact  fo| 
Common  practice  :  which  \vill  fie  found  very  convenient. 


AH'KNDIX. 


219 


TABLE  of  Breadth,  Height,  and  (Content, 


[  Breadth. 

Height  infect 

-. 

[ft.     in. 

I 

2 

3 

4 

1    l 

i  910  n" 

2-   6 

15 

3C 

iH 

60 

I  — 
1 

21  4!  5    6i  7  7)  l(j 

1 

tni2 

14 

7 

16 

3147 

62 

1 

3'i  4i  5!  { 

8    9 

10 

12 

J13 

14 

8 

16 

32 

4864 

1    3|  4i  5i  7 

8    9 

11 

12 

13  15; 

9 

17 

33 

4966 

I    « 

4    C> 

7 

8 

9 

11 

12 

14 

15! 

10 

17 

345168,    2 

S 

4    6 

7 

£ 

10 

11 

13 

114 

11 

18 

3553 

70 

2 

3,  4,'  6    7 

9 

>w 

13 

13115 

3    0 

18 

3654 

72 

!2 

8 

5!  i) 

8|  9  11112 

14 

15 

if] 

1  ' 

19 

37(56  74 

2 

3 

5j  6 

8 

911 

12 

14 

16 

17 

2 

19,  *8J57 

76 

o 

3 

5i  6 

8 

ion 

13 

14 

1617' 

3 

1939:59 

78 

2 

f> 

5;  7   8 

10111 

1315 

16  JH 

4 

20 

40 

60 

80 

o 

g 

5    7 

8 

10:1213 

1517  I:- 

5 

21  41  62 

82 

2 

Q 

ft*    * 

iDlial 

14 

16117  l!>; 

"IT 

21 

4263 

Si 

2 

4 

5    7 

^  i 

11 

12 

u! 

I9i 

7 

22143  64 

§6 

2 

4 

"q  7  9; 

11 

13114 

L6 

1820' 

22 

14  66,88| 

AW 

4 

61  7 

9 

u 

13J15 

17 

I82QJ 

9  |J23< 

45  081901 

2 

| 

11 

13]] 

15' 

17 

10  Ii23< 

16 

69192 

4  4J  (3    7j  o; 

12 

l;j!ir>  j;  r 

11  jfegfc 

17 

r094 

2.1  4i  ei  Slid! 

13  I4|l6 

L820J 

4     0  !|24|48  72*96 

2'  4l  (>!  8 

; 

!  >  i4il 

6 

TO  USE  THE  FOREGOING  TAS.U^. 

First  measure  the  breadth  and  heisht  of  voui-Ioad  to  (he  ii! 
in?,h  ;  then  find  the  breadth  in  the  left  hand  column  pi' the  table,  then  mov 
to  the  right  on  the  same  line  till  you  come  under  the  height  in  feet,  and  you 
will  have  the  content  in  inches,  answering  the  feet,  to  which  #»ld  ih^coaieir. 
of  tlie  inches  on  the  riffht  and  divide  the  sum  by  12,  and  you  will  . 
true  conteni  of  the  load  in  feet  and  inches. 

»/VoJ€. — The  contents  answering  the  inches  btMiig  nhvays  smfQi^may  t  • 
added  by  inspection. 

EXAMPLES, 

1.  Admit  a  load  of  wood  is  3  feet  4  inches  wide,  and  2  feet  10  inches  liigh, 
required  the  content. — 

Thus,  against  3  fe«t  4  inches,  and  under  2  feet,  stands  40  inches  ;  and  tin- 
der 10  inches  at  top,  stands  17  inches:  then  40-H?=57,  true  content  in 
inches,  which  divide  by  12,  ffivos  4  feet  9  inches,  the  answer. 

55.  The  breadth  being  3  feel,  and  height  -2  fret  8  inches;  required  the  con 
tent.— 

Thti«,  with  brr-ad'li^  H-'  °  •'    ''°P-  sjftnas  36 


220 


APPENDIX. 


inches  ;  and  under  8  inches,  stands  12  inches  :  now  36  and  12  make  48,  the 
answer  in  inches  ;  and  48-^12=4  feet,  or  just  half  a  cord. 

3.  Admit  the  breadth  to  be  3  feet  11  inches,  and  height  3  feet  9  inches  ; 
required  the  content. 

Under  3  feet  at  top,  stands  70 ;  and  under  9  inches,  is  18  :  70  and  18,  make 
S8-M2=7  feet 4  inches,  or  7  ft.  1  qr.  2 inches,  the  answer. 


TABLE  I. 


Showing  the  amount  of  £1,  or  $1,  at  5  and  6  pe\ 
annum,  Compound  Interest,  for  20  year. 


5  and  6  per  cent,  per 


Yrs. 

5  per  cent.'C)  per  cent.  \\  Yrs. 

5  per  cent. 

6  per  cent. 

I 

1  ,05000 

1,05000 

11 

1,71034 

1,89829 

0 

1,10250 

1,12360 

12 

1  .79585 

2,01219 

'3 

1,1571)2 

1,19101  i  13 

1,88565 

2,13292 

4 

1,21550 

,26247 

i  14 

1,97993 

2,26090 

;> 

1,27628 

,33322 

15 

2,07893 

2,39655 

<> 

1,34009 

,41851 

16 

2,18287 

2,54727 

7 

1,40710 

,50363 

17 

2,29201 

2,69277 

8 

1,47745 

,59384 

18 

2,40661 

2,85433 

9 

1,55132 

1,68947 

19 

2,52695 

3,02559 

10 

1,62889 

1,79084 

20 

2,65329 

3,20713 

VII.    The  weights  of  the  coins  of  the  United  States. 


Eagles, 

Half-Eagles, 

Quarter- Eagles, 

Dollars, 

Half-Dollars, 

Quarter-Doll  ars , 

Dimes, 

Half-Dimes, 

Cents, 

Half-Cents, 


•  pwt.    grs. 
11       6 


2 

17 

8 
4 
1 

8 
4 


16 

8 
171 

20i 
16 

8 


Standard 
Gold. 


Standard 
Silver. 


Copper. 


The  standard  for  gold  coin  is  11  parts  pure  gold,  and 
one  part  alloy — the  alloy  to  consist  of  silver  and  copper. 
The  standard  for  silver  coin  is  1485  part's  fine  to  179  parts 
alloy — -the  nllov  to  bo  \vhollv  copper. 


ANNUITIES. 


TABLE  il.             ||        TABLE  iil. 

Slioioing  the  amount  of  £\  anmn-\  Showing  the  present  worth] 
/y,  forborne,  fcr  31  year."  or  H:i-\\     of  £1  cnnrdty^  to  conli\ 
aer,  at  5  and  6  per  cent,  cwft-      nvc  for  $1  years,  at  !>  afld 
pound  interest.                             \\     8  per  cent.  compoundifU.\ 

Yrs.           5 

6                  -3                 6       j 

1 
2 
3 
4 
5 

1,000000 
2,050000| 
3,152500 
4,310125 

5,525681! 

1,000000:    0.952381 
2,(^0000|    1,859410 
3,183fm    ^,723248 
4,3740  JG     8,'545950 
5,60710:)     4  ,329477 

0,943396 
1,833398 

2,673012; 
3,165106] 
4,212361] 

G 

z 

9 

Ji. 
11 
10 

LA 

13 
14 
15 

0,801913 
8,142009 
9,549109 

11,026564 

12,577892 

6,975319     5,07509^ 
8.^933"^     5,786278 
9,£974(>S    6,463213 
11,491310    7,10 
13,180?  '0!    7,7^17:35 

4,917324; 
5,582381 
6,2097941 
6,8016921 

^087; 

14,206787 
15,917126 
17,712982 

19,598C32 
21,578564 

14,971643] 
16,86994% 

18,882138 
21,015 

23,275%t) 

8,306414 
8,G63252 
9,393573 
9,898641 

10,379668 

7JS& 

8,:^3844 
8,852683' 
932919S4' 
9,712249; 

16 
17 
18 
19 
20 

23,<>57492  25,67#>23 
25,840366  28,2123801 
28,13238530,905653 
30,539004  33,'/o99^ 
33,065954  36,785592] 

10,837769 
1I.2740C6 
11,689587 
13,085321 
1:2,462210 

10,105895. 
10,477-260! 
10,827tJ03i 
11,158116. 
11,469921! 

21 
22 
23 
24 
25 

35,719252 
38,505'2l4 
41,430475 
44,501999 

47,727099 

39,992727 
43,39-2291 

46,995828 
50,815578! 
54,864512 

12,821153 
13,163003 

13,488574 
13,798642 
14,093944 

11,764077} 
12,041582; 
12,303380; 
12,550357] 

12,783356 

< 

26 
27 
28 
29 
30 
31 

51,113454 
54,669126 
58,402583 
62,322712 
66,438847 
70,760790 

59,1,5638-2 
63,705765 
68,52811-2 
73,639798 
179,058186 
184,801677 

14,375185 
14,613034 
14,898127 
15,141073 
15,372451 
15,50-2810 

13,003166^ 
13,210534 
13,10(5164 
13,590721! 
13,764831 
13,929086 

TABLES. 


THE  three  following  tables  are  calculated  agreeable  to 
an  Act  of  Congress  passed  in  November,  171^2,  making 
foreign  Gold  and  Silver  coins  a  legal  tender  for  the  pay* 
ment  of  all  debts  and  demands,  at  the  several  and  respec-  - 
live  rates  following,  viz.  The  Gold  Coins  of  Great  Bri- 
tain and  Portugal,  of  their  present  standard,  at  the  rate  of 
100  cents  for  every  27  grains  of  the  actual  weight  there- 
of.— Those  of  France  and  Spain  27f  grains  of  the  actual 
weight  thereof. — Spanish  milled  dollars  weighing  17  pwt. 
7  gr.  equal  to  100  cents,  and  in  proportion  for  the  parts  of 
a  dollar. — Crowns  of  France  weighing  18  pwt.  17  gr. 
equal  to  110  cents,  and  in  proportion  for  the  parts  of  a 
Crown. — They  have  enacted,  that  every  cent  shall  contain 
208  grains  of  copper,  and  every  half-cent  104  grains. 

TABLE  IV. 


Weights  of  several  pieces  of  English,  Portuguese  and 
French  Gold  Corns. 


|  Pwt. 

|    Gr. 

\Dols.   Cts.     M. 

Johannes,    -  -  -  -  - 

18 

16       0       0 

Single  ditto,   -  -  -  - 

9 

800 

English  Guinea,  - 

5 

G 

4     66| 

Half       ditto,     

2 

15 

2     33i 

French  Guinea,  -  -  - 

5 

6 

4     59      8 

Half       ditto.     

2 

15 

2    29      9 

4  Pistoles,    

16 

12 

14     45      2 

2  Pistoles,    

8 

6 

7    22       6 

1  Pistole,     

4 

3 

3    61       3 

oo 

*& 

6     14      8  j 

APPENDIX. 


-5  §  1 


^D^CO       CO  ^J  Oi  »O  W  QO  »O  r-H  QO  rj<  rH  CO  rj»  O  ts.  TO       CC  JO  Oi 


a" 


-42 


SCO*?! 


4  <M  CO  *f  Id  O  is.  00  O>  O  r-i  <M  CO  rt<  IO  CO  Is.  OO  Ci  yr-lC^CC 


AFPENiMX, 


VII.    TABLE    of  Cents,  answering  to   the    Currencies 

of  the  United  States,  with  Sterling,  #c. 
NOTE. — The   figures  on  the  right    hand    of  the  space,, 
show  the  parts  of  a  cent,  or  mills,  &c. 


p. 

6s.   to 
the 
Doll. 

8s.   to 
the 
Doll. 

Is.tid. 
to  the 
Doll. 

to  the 
Doll. 

5s.   to 

the 
Dell. 

4*.6d. 
tor  the 
Doll. 

*    thl 
Dollar. 

cents. 

cents. 

cents. 

cents. 

cents. 

cents. 

cents. 

I 

I  3 

1  0 

I    1 

1  7 

16      18 

1     7 

o 

2  7 

2  0 

2  2 

3  5 

3  3 

3  7 

3     4 

3      4  1 

3  ll     3  3 

5  3 

5 

5  5 

5     1 

4l     5  5 

4  1 

4  4 

7  1 

63 

7  4 

6     8 

5 

6  9 

5  2 

5  5|     8  9 

8  6 

9  2 

8    5 

C 

8  3 

6  2 

6  6    10  7 

10 

11   1 

10    2 

7 

9  7 

7  2 

7  rl   12  5 

11   (> 

12  9 

11     9 

8 

11   1 

8  3 

8-8 

14  2 

13  3 

14  8 

13     6 

9    12  5 

•    9  3 

10 

16 

15 

16  6 

15    3 

10]  13  8 

10  4 

^ll  "l 

17  8 

16  6 

18  5 

17 

11 

15  2 

11    1 

12  2 

19  0    18  3 

20  3 

18 

8. 

i 

1 

10  6 

12  5 

13  3j  21  4    20 

22  2 

20 

k     2 

33  3 

25 

26  0 

42  8    40 

44  4 

41 

3 

50 

37  5 

40 

61  2 

60 

66  6 

61     5 

4|  GO  6 

50 

53  3 

85  7 

80 

88  8 

82 

5   83  3 

62  5 

66  6 

107  1 

100 

111   1 

102     5 

6 

100 

75 

60 

128  5 

120 

133  3 

123 

ly 

/ 

116  6 

87  5 

93  3 

150 

140 

155  5 

143     5 

8133  3 

100      106  6 

171  4 

160      177  7 

164     1 

9J160 

112  5 

120 

193.8180     200 

184     6 

10  166  6 

125 

133  3 

214  2  200      222  2 

205     1 

11  183  3 

137  5 

146  6235  7220     |244  4 

225    6 

12 

200 

150 

160     257  1  240 

266  6 

246     1 

13 

216  6 

162  5 

173  3  278  5 

260 

288  8 

266    6 

14 

233  3 

175 

186  6 

300 

2SO 

311  1 

287     1 

15 

250 

187  5 

200 

321  4 

300 

333  3 

307    6 

16 

266  61200 

213  3 

342  8 

320 

355  5 

328    2 

17 

283  3 

212  5 

226  6 

364  2 

340 

377  7 

348    7 

18 

300 

225 

240 

385  6 

360 

400 

369    2 

19 

316  6 

237  5 

253  3 

407  1 

380 

422  2 

389    T 

20 

333  3 

250 

266  6 

428  5 

400 

444  4 

410     2 

APPENDIX. 


TABLE  IX. 


Shewing  the  value  of  Federal  Money  in  other  Currencies. 


Federal 

Money. 

New  Eng- 
land i    Vir- 
ginia ,    and 
Kentuky 
currency. 

New    York 
and  North 
Carolina 
currency. 

New  Jersey, 
Pennsylva- 
nia ,     Dela- 
ware, and 
Maryland 
currency. 

South-Car-] 
olina,    and  | 
Georgia 
currency. 

Cents. 

s.     d. 

s.     d. 

s.     d. 

s.      d. 

1 

0     OJ 

0     1 

0     1 

0     Oi 

2 

o    11 

0     2 

0     1J- 

o  r 

3 

0    2i 

0     3 

0    2J 

0     If 

4 

0     3 

0     3J 

0     3L 

0    21 

5 

0     3i 

0     4f 

0     4i 

0    2J 

6 

0     4i 

0     5£ 

0     5i 

0    3| 

7 

0     5- 

0     6£ 

0     61 

0     4 

8 

0     5? 

0     7| 

0     7J 

0    4i 

9 

0     61 

0    8J 

0    8 

0     5 

10 

0     7i 

0     9i 

0     9 

0     5} 

11 

0    8 

0  101 

0  10 

0    6} 

12 

0    8J 

0  Hi 

0  10J 

0    6? 

13 

0     9i 

1     Oi 

0  11  J 

0     7* 

14 

0  10 

1  H 

1     0V 

0     7f 

15 

0  10J 

1     2i 

i    H 

0  -8i 

16 

0  Hi 

3i 

1     0£ 

0     9 

17 

1     Oi 

4£ 

1   ?i 

0     9i 

18 

1    1 

51 

1     4i 

0  10- 

19 

1  If 

6i 

1     5} 

0  10  J 

20 

1     2i 

W 

1     6 

0  11} 

,       30 

1     9i 

2     4J 

2     3 

1     4J 

40 

2     4J 

3     2>V 

3     0 

1   10i 

50 

3     0 

4     0~ 

'       3     9 

2     4" 

60 

3     7i 

4     9^ 

4    6 

2    9£ 

70 

4    2i 

5     7i 

5     3 

3     3} 

80 

4    9i 

i\     4^ 

6     0 

3     8f 

90 

5     4f 

7     2i 

6     9 

4    2.V 

100 

6     0 

>       8     0 

7     6 

4     8" 

226  APPENDIX. 

A  FEW  USEFUL    FORMS  IN    TRANSACTING    BUSINESS. 

AN  OBLIGATORY  BOND. 

KNOW    all    men  by    these    presents,  that    I,  C.  D.  of 
in  the  county  of  am  held  and    firmly  bound   to 

II.  W.  of  in  the  penal  sum  of  to  be  paid 

II.  W.  his  certain  attorney,  executors,  and  administrators ; 
to  which  payment,  well  and  truly  to  be  made  and  done, 
I  bind  myself,  my  heirs,  executors,  and  administrators, 
firmly  by  these  presents.  Signed  with  my  hand,  and 
sealed  with  my  seal.  Dated  at  this  day 

of  A.  D. 

The  condition  of  this  obligation  is  <fuch.    That    if  the 
above  bounden  C.    D.    &c.    [Here    insert    the   condition^ 
then  this    obligation  to  be  void  and  of  none  effect ;  other- 
wise to  remain  in  full  force  and  virtue. 
Signed,  scaled,  and  delivered,  [ 
in  the  presence  of  J 

A  BILL  OF  SALE. 

KNOW  all  men  by  th^se  presents,  that  I,  B.  A.  of 
for  and  in  consideration  of  to  me  in  hand   pard  by 

D.  C.  of  the    receipt  whereof  I    do    hereby  ac-4 

knowledge,  have  bargained,  sold,  and  delivered,  and,  by 
these  presents,  do  bargain,  sell  and  deliver  unto  the  said, 
D.  C.  [Here  specify  the  property  sold.]  To  HAVE  and  to 
HOLD  the  aforesaid  bargained  premises,  unto  the  said  D.  C. 
his  executors,  administrators,  and  assigns,  forever.  And  I 
the  paid  B.  A.  for  myself,  my  executors  and  administrators, 
shall  and  will  warrant  and  defend  the  same  against  nil  per- 
sons unto  tfce  said  D.  C.  his  executors,  administrators,  and 
assigns,  by  these  presents.  In  witness  whereof,  I  hnvo 
hereunto  set  my  hand  and  seal,  this  day  of  1814. 
In  presence  of 

A  SHORT  WILL. 

I,  B.  A.  of,  &c.  do  make  and  ordain   this    my  last  will 
flnd  testament,  in  manner  nnr!  form  following,   vi/.  I 


Al'FENDJX. 

uud  bequeath  tu  my  deaf  brother,  R.  A.  the  sum  of  ten 
pounds,  to  buy  him  mourning.  I  give  and  bequeath  to 
my  son  J.  A.  the  sum  of  two  hundred  pounds.  I  give  and 
bequeath  to  my  daughter  E.  E.  the  sum  of  one  hundred 
pounds  ;  and  to  my  daughter  A.  V.  the  like  sum  of  one 
hundred  pounds.  All  the  rest  and  residue  of  my  estate, 
goods  and  chattels,  1  give  and  bequeath  to  my  dear  lie- 
loved  wife,  E.  R.  whom  I  nominate,  constitute  and  appoint 
sole  executrix  of  this  my  last  will  and  testament,  hereby 
revoking  all  other  and  former  wills  by  me  at  any  time 
heretofore  made.  In  witness  whereof,  I  have  hereunto 
set  my  hand  and  seal,  the  day  of 

in  the  year  of  our  Lord 

Signed,  sealed,  published  arid  declared  by  the  said  tes- 
tator, B.  A.  as  and  for  his  last  will  and  testament,  in  the 
presence  of  us  who  have  subscribed  our  names  as  witnesses 
thereto,  in  the  presence  of  the  said  testator. 

R.  A. 
S.  D. 
L.  T. 

NOTE. — The  testator,  after  taking  off  his  seal,  must,  in 
presence  of  the  witnesses,  pronounce  these  words :  "  I 
publish  and  declare  this  to  be  my  last  will  and  testament." 

Where  real  estate  is  devised,  three  witnesses  are  ab- 
solutely necessary,  who  must  sign  it  in  the  presence  ol' 
the  testator. 


A  LEASE  OF  A  HOUSE. 
KNOW  all  men  by  these  presents,  that  I,  A.  B.  of 
in  for  and  in  consideration  of  the  sum  of  re- 

ceived to  my  full  satisfaction  of  P.  V.  of  this 

day  of  in  the  year  of  our  Lord  have  demised 

and  to  farm  let,  and  dp  by  these  presents,  demise  and  to  farm  let, 
unto  this  said  P.  V.  his  heira,  executors,  administrators  and  as- 
signs, one  certain  piece  of  land,  lying  and  being  situated  in  said 
bounded,  &c.  [Mere  describe  the  boundaries]  with  a 
dwelling  house  thereon  standing,  for  the  term  of  one  year  from 
this  date.  To  HAVE  and  to  HOLU  to  him  the  said  P.  V.  his  heirs, 
executors,  administrators  and  assigns,  for  said  term,  for  him  the 
said  P.  V.  to  use  and  occupy,  as  to  him  shall  seem  meet  and 
proper.  And  the  said  A.  JB.  doth  FURTHER  COVENANT  with  the 


228  APPENDIX. 

said  1\  that  he  huth  good  right  to  let  and  demise  the  said 
letten  and  demised  premises  in  manner  aforesaid,  and  that  he> 
the  said  A.  during  the  said  time  will  suffer  the  said  P.  quietly  to 
HAVE  and  to  HOLD,  use,  occupy  and  enjoy  said  demised  premises, 
and  that  said  P.  shall  have,  hold,  use,  occupy,  possess. and  enjoy 
the  same,  free  and  clear  of  all  incumbrances,  claims,  rights  arid 
titles  whatsoever.  In  witness  whereof,  I  the  said  A.  B.  have 
hereunto  set  my  hand  and  seal,  this  day  of 

Signed,  sealed  and  delivered  > 

in  presence  of  $  A.  15. 

A  NOTE  PAYABLE  AT  A  BANK. 

#500,60]  HARTFORD,  May  30,  1815. 

FOR  value  received,  I  promise  to  pay  to  John  Merchant, 
or  order,  Five  Hundred  Dollars  and  Sixty  Cents,  at  Hartford 
Bank,  in  sixty  days  from  the  date. 

WILLIAM  DISCOUNT. 

AN  INLAND  BILL  OF  EXCHANGE. 
[$83,34]  BOSTON,  June  1,  1815. 

TWENTY  days  after  date,  please  to  pay  to  Thomas  Good- 
win or  order,  Eighty-Three  Dollars  and  Thirty-Four  Cents,  and 
place  it  to  my  account,  as  per  advice  from  your  humhle    servant, 
Mr.  T.  W. Merchant,  \  SIMON  PURSE. 

New-  York. 


A  COMMON  NOTE  OF  HAND. 

[#1301  NEW-YORK,  March  8,  1821. 

FOR  value  received,  I  promise  to  pay  to  John  Murray,  One 
Hundred  and  Thirty  Dollars,  in  four  months  from  this  date,  with 
interest  until  paid.  JOHN  LAWRENCE. 


A  COMMON  ORDER. 

NKW-YORK,  June  10, 
Mr.  Charles  Careful, 

Please  to  deliver  Mr.  George  Speedwell,  the  amount  of 
Twenty-Five  Dollars,  in  goods  from  your  store ;  and  charge  the 
same  to  the  account  of  Your  Ob't.  Servant, 

E.  WHITE, 


FINIS. 


THE 

PRACTICAL   ACCOUNTANT, 

OR, 

FARMERS'  AND  MECHAN1GKS' 

BEST    METHOD  OF 

BOOK-KEEPING; 

FOR  THE 

EASY  INSTRUCTION  OF  YOUTH. 


DESIGNED   AS 


A   COMPANION 


DABOLL'S  ARITHMETICS 


BY  SAMUEL  GREEN. 


M1DDLETOWN,  (Con.) 

PUBLISHED  BY  WILLIAM    H.  NILES. 

Stereotyped  by  A.  Chandler,  New- York. 

18-28. 


INTRODUCTION. 


SCHOLARS,  male  and  female,  after  they  have  acquired  a  sufficient 
knowledge  of  Arithmetic,  especially  in  the  fundamental  rules  of  Addi- 
tion, Subtraction,  Multiplication,  and  Division,  should  be  instructed 
in  the  practice  of  Book  Keeping.  By  this  it  is  not  meant  to  recom- 
mend that  the  son  or  daughter  of  every  farmer,  mechanic,  or  shop 
keeper,  should  enter  deeply  into  the  science  as  practised  by  the  mer- 
chant engaged  in  extensive  business,  for  such  studynvould  engross  a 
grt<it  portion  of  time  which  might  be  more  usefully  employed  in  ac- 
quiring a  proper  knowledge  of  a  trade,  or  other  employment. 

Persons  employed  in  the  common  business  of  life,  who  do  not  keep 
regular  accounts,  aro  subjected  to  many  losses  and  inconveniences  ; 
to  avoid  which,  the  following  simple  and  correct  plan  is  recommend- 
ed for  their  adoption. 

Let  a  small  book  be  made,  or  a  few  sheets  of  paper  sewed  toge- 
ther, and  ruled  after  the  examples  given  in  this  system.  In  the  book, 
termed  the  Day  Book,  are  duly  to  be  entered,  daily,  all  the  transac- 
tions of  the  master  or  mistress  of  the  family,  which  require  a  charge 
to  be  made,  or  a  credit  to  bo  given  to  any  person,,  No  article  thus 
subject  to  be  entered,  should  on  any  consideration  be  deferred  till 
another  day.  Great  attention  should  be  given  to  write  the  transac- 
tion in  a  plain  hand  ;  the  entry  should  mantion  all  the  particulars  ne- 
cessary to  make  it  fully  understood,  with  the  time  whei-  they  took 
place ;  ^nd  if  an  article  be  delivered,  the  name  of  the  person  to  whom 
delivered  is  to  be  mentioned.  No  scratching  out  may  be  suffered  ;  be- 
cause it  is  sometimes  done  for  dishonest  purposes,  and  will  weaken 
or  destroy  the  authority  of  your  accounts.  --But  if,  through  mistake, 
any  transaction  should  be  wrongly  entered,  the  error  must  be  rectified 
by  a  new  entry  ;  and  the  wrong  one  may  be  cancelled  by  writing  the 
word  Error  in  the  margin. 

A  book,  thus  fairly  kept,  will  at  all  times  show  the  exact  state  of  a 
persons  affairs,  and  have  great  weight,  should  there  at  any  time  be  a 
necessity  of  producing  it  in  a  court  of  justice. 


FORM  OF  A  DAY  BOOK. 


MEREMIAH  GOOD  ALE,  Albany,  January  1,  1822. 


Entered. 
1 

Entered. 
1 

Entered. 
1 

Entered. 
1 

Entered. 
1 

Entered. 
1 

Entered. 
1 

Entered. 
1 

Entered. 
1 

Joseph  Hastings,                                               Cr. 
By  3  months'  wages,  at  $6  a  month,  due  this  date, 
5                       • 

?8 

1 

11 
11 

100 

5 

74 

3 
2 

ct. 

00 

50 

50 
50 

00 
£4 

30 

62 
16 

Samuel  Stacy,                                                  Dr. 
To  2  weeks'  wages  of  my  daughter  Ann,  spinning 
yarn,  at  75  cents  a  week,  ending  this  day, 

Joseph  Hastings,                                             Dr. 
To  my  order  for  goods  out  of  the  store  of  Anthony 
Billings,       ....... 

Anthony  Billings,                                             Cr. 
By  my  order  in  favour  of  Joseph  Hastings, 
15 

Thomas  Grosvenor,                                        Dr. 
To  the  frame  of  a  house  completed  and  raised  this 
day  on  his  Glover  Farm,  so  called,  4000  feet  at  2i 
cents  per  foot,      ...... 

10 

Edward  Jones,                                                 Cr. 
By  his  team  at  sundry  times,  carrying  manure  on 
my  farm,      

-  -     _,              ^5 

Thomaw  Grosvenor,                                        Dr. 
To  48  window  sashes  delivered  at  his  Glover  Farm, 
so  called,  at  $1,00      .         .    .     .              $48,00 
Setting  500  panes  of  glass  by  my  son  John, 
at  lj  cents,         7,50 

10  days'  work  of  myself  finishing  front  a-oom, 
at  $1,25  a  day,         ....          12,50 
7?  do.  of  William,  my  hired  man,  laying  J 
the  kitchen  tiuor  and  hanging  doors,  at  >  6,30 
84  cents  a  4*y?                                              S  
:            ^6  ••  ---i 

Anthony  Billings,       .     »                                   Cr. 
By  2  galls,  molasses,  at  36  cts.  per  gall.              0,72 
4  yds.  of  India  Cotton,  at  ISA  cents,                0,74 
2  flannel  shirts  to  Joseph  Hastings,                 2,16 

Joseph  Hastings,                                              Dr. 
To  2  shirts  of  A.  Billings,        .... 

There  put  the  name  of  the  owner  of  the  book,  and  first  date, 


FORM  OF  A  DAY  BOOK. 


Albany,  February  12,  1822. 


Entered. 
t 

Thomas  Grosvenor,                                           Cr. 
By  my  order  in  favour  of  Joseph  Hastings, 

Entered. 
1 

Joseph  Hastings,                                               Dr. 
To  my  order  on  T.  Grosvenor, 
16 

Entered. 
1 

Thomas  Grosvenor,                                          Dr. 
To  3  days'  work  of  myself  on  your  fence  at  $1,25 
per  day,     3,75 
3  days'  do.  my  man  Wm.  on  your  stable  and 
finishing  off  kitchen,  at  84  cts.   .         .         2,52 
2  pr.  brown  yarn  stockings,  at  42  cts.             0,84 

18 

Entered. 

1 

Edward  Jones,                                                   Cr. 
By  4  months'  hire  of  his  son  William  at  $  10  a  month, 

OA 

Entered. 
1 

Edward  Jones,                                                   Dr. 
To  my  draft  on  Thomas  Grosvenor, 

Entered. 
1 

•  Thomas  Grosvenor,                                     _  Cr. 
By  my  draft  in  favour  of  E.  Jones, 

00 

Entered. 
1 

Thomas  Grosvenor,                                          Dr. 
To  the  frame  of  a  barn,            .... 

Entered. 
1 

Anthony  Billings,                                            Cr; 
For  the  following1  articles, 
14  Ibs.  muscovado  sugar  at  $12  pr  cwt.          1,50 
1  large  dish,           .....         0,23 
6  plates,         0,30 
4  cups  and  saucers         ....         0,20 
1  pint  French  Brandy,             .         .         .         0,17 
1  quart  Cherry  Bounce,           .          .          .         0,33 
Thread  and  tape,            ....         0,18 
2  Thimbles,            0,04 
1  pair  Scissors,       .         .         .         .         ,         0,17 

Wafers,  4  ;  ink,  6  ;  1  bottle,  8  ;         .         .         0,18 

Entered. 

1 

Peter  Daboll,                                                    Dr.     1 
To  a  cotton  Coverlet  delivered  Sarah  Bradford,  byfj 
your  written  order,  dated  14  Jan, 

355 


151 


FORM  OF  A  DAY  BOOK. 


Albany,  March  1, 


Entered. 
1 

Entered. 
1 


Entered. 
1 


Entered. 
1 

Entered. 

1 


Thomas  Grosvenor, 
By  cash  paid  me  this  date, 


Cr. 


Anthony  Billings,  Dr. 

To  one  barrel  of  Cider,    .         .         .         .$11 
1  barrel  containing-  the  same,  (from  Tho- 
mas Grosvenor,)        .         .         .         .         0  58 


7_ 


Thomas  Grosvenor,  Cr. 

By  1  barrel  containing  Cider  sold  and  delivered  to 
Anthony  Billings,     . 

10 


Anthony  Billings, 
To  cash  per  his  order  to  George  Gilbert, 
— 1 5 


Dr. 


Peter  Daboll,  Cr. 

By  amount  of  his  Shoe  account,       .         .       $448 
Yarn  received  from  him  for  the  balance  of 
his  account,       .....         1 


Entered.      Samuel  Green,  Cr. 

2  By  amount  due  for  12  months  New-London 

Gazette,  $2  00 

4  Spelling  Books,  at  20  cents,  for  children,  0  80 

1  Daboll's  Arithmetic,  for  my  son  Samuel,     0  42 

2  blank  Writing  Books,  at  124  cents,     .         0  25 
1  quire  of  Letter  Paper,         .         .         .         034 


Entered, 


Entered 


Notes  Payable,  Dr. 

2  By  my   note  of  this  date,   endorsed  by  Ephraim 
Dodge,  at  6  months,  fbr  a  yoke  of  Oxen  bought 
of  Daniel  Mason,  at  Lebanon, 
.28 


-24- 


Jonathan  Curtis,  Dr. 

2  To  an  old  bay  Horse,         ....     #23  00 
A  four  wheeled  Wagon,   and   half  worn 

Harness,  .         .         .         .         .       42  00 


Entered.      Samuel  Green, 
2  To  cash  in  full, 


Dr. 


FORM  OF  A  DAY   BOOK. 


Albany,  April  6, 


Entered.      Anthony  Billings, 

To  2  tons  of  Hay,  at  $1 1   25, 


Entered. 
1 


Entered. 
1 


D,r. 

.         .     $2250 
Amount  of  order  dated  March  26,  1822,  ) 

inTavour  of  Fanny  White,  paid  in  1  >      0  54 
pair  yarn  stockings,  .         .          ) 

Hire  of  my  wagon  and  horse  to  bring 

sundry  articles  from   Providence,  3d         3  00 
of  this  month,    .         .         .         . 


Thomas  Grosvenor, 


-12- 


Cr. 


By  his  order  on  Theodore  Barrel],  New-London,  for 
68  dollars,         .         .        ..... 


Ajithony  Billings,  Dr. 

To  1  hogshead  Rum  from  Theodore  Barrel!, 

100  gals,  at  50  cents,         .         .         .     $50  00 
Cash  received  from  said  Barrell  for  balance 

due  on  Thomas  Grosvenor's  order,  18  00 


Entered. 


Entered. 

i. 


Entered. 


Jonathan  Curtis, 


-18- 


Cr. 


2  By  a  coat  $14,75,  pantaloons  $5,00, 

-22 


Thomas  Grosvenor,  Dr. 

To  mending  your  cart  by  my  man  William,    $1  00 
Paid  Hunt  for  blacksmith's  work  en  jrour 

cart,  .         .         .         .         .         .         0  58JJ 

Setting  6  panes  of  glass,  and  6nding  glass,    0  66 


2  To  a  yoke  of  Oxen,  at  60  days'  credit 
Entered. 


Entered. 


-25- 


John  Rogers, 


Dr. 


Anthony  Billings,  Cr. 

By  Garden  Seeds  of  various  kinds,     .         .     $0  56 
1  pair  Boots,  myself,  $4,00,  and  1  pair  for 

John,  f  3,50 7  50 

1  pair  of  thick  Shoes  for  Joseph  Hastings,     1  25 
Tea,  Sugar,  and  Lamp  Oil,  per  bill,       .       0  68 


Notes  Payable,  Cr. 

By  my  note  to  Isaac  Thompson,  at.  &  months, 


FORM  OF  A  DAY  BOOK. 


Albany,  May  3, 


Entered. 

2 

Entered. 
I 

Entered. 
1 

Entered. 
1 

Entered. 

2 

Entered: 
1 

_ 

Entered. 
1 

Entered. 

2 

Theodore  Barrel!,  New-London," 
To  16,  cheese,  308  Ibs.  at  5  cents,      . 
217  Ibs.  of  butter,  at  15  2-3  cts.    . 
24  Ibs.  of  honey,  at  12$  cents, 

<] 

Dr.     1 
$15  40! 
34  00, 
3  00 

$ 

52 
1 

43 
31 

52 

54 

54 
48 

ct. 

40 

25 

60 
50 
40 

00 

00 
00 

Joseph  Hastings, 
To  1  pair  shoes,  29lh  April,  from  Anthony 
1° 

Dr. 

Bilikigs, 

Anthony  Billings,                                             Dr. 
To  84  bushels  of  seed  potatoes,  at  33  1-3 
cents,        .         .        .        ,".;-.         .     $28  00 
8  pair  mittens,  at  20  cents,   .         .         .          1  60 
Cash,            .         .         .'                 .                 14  00 

15 

Joseph  Hasting^, 

By  4£  months  wages,  at  7  dollars, 
20 

Cr. 

Theodore  Barrel!, 
By  cash  in  full  of  all  demands, 

°5 

Cr. 

Thomas  Grosvenor, 
By  his  acceptance  of  my  order  in  favour  .of 
Billings,              ..... 

Cr. 

Anthony 

Anthony  Billings,                                              Dr. 
To  amount  of  my  order  on  Thomas  Grosvenor, 

C~p£     r>£ 

Notes  Payable, 
To  cash  paid  for  my  note  to  D.  Mason, 

Dr. 

Tho  foregoing  example  of  a  Day  Book,  may  suffice  to  give  a  good  idea  of  the  way 
in  which  it  is  proper  to  make  the  original  eu'.nVs  of  all  dent  and  credit  articles.  Ano- 
ther small  book  should  mx?  bo  ur;*pa;v.d.  according  to  the  following  form,  termed  tiu» 
book  of  Accounts,  or  Leger.  Into  this  book  must  be  posted  the  whole  contents  of  the 
Day  Book  ;  care  being  taken  that  every  article  be  carried  to  its  corresponding  title  ; 
the  debt  a-.nomUs  to  b-  -•  ;r---^3  l.\  th*-  left,  aiul  (  .  right  hand  p::gr.  i'hus, 

should  it  at  any  time  be  required  to  know  the  state  of  nn  account,  it  will  only  be  neces- 
sary to  sum  up  the  two  columns,  and  to  subtract  the  smaller  amount  from  the  greater, 
the  remainder  will  be  the  balance. 

When  an  article  is  posted  from  the  Day  Book  into  tiie  Leger,  it  will  i>e  proper,  op. 
poaite  the  article,  to  not '  the  same  in  the  margin  of  the  Day  Book,  by  writing  th#  word 
Entered,  or  making  two  parallel  strokes  with  the-  pen  :  to  which  should  be  added  the 
figure  denoting  the  pa,^e  m  the  Leger  where  the  account  is. 

On  a  bi^nk  pag*  at  the  beginning  or  cad  of  the  Leger,  an  alpnabetical  index  should 
be  written,  containing  the  names  of  every  person  with  whom  you  haveacc.ount^in  the 
Leger,  with  the  number  of  the  pace  where  the  accounts  are. 


FORM  Or  A  LEGER. 


Dr. 


Joseph  Hastings. 


1822. 
Jan'y 

Feb'y 
May 

5 
26 
12 
8 

To  my  order  on  Anthony  Billings  for  goods, 
2  shirts  of  Anthony  Billings, 
My  order  on  Thomas  Grosvenor, 
1  pair  shoes.  29th  ^pril,  from  A.  Billings,     - 

fa 

I 

1 

Ct. 

50 
16 
50 
25 

Dr. 

Samuel  Stacy. 

1822. 
Jan'y 

5 

To  2  weeks'  wages  of  my  daughter,  at  75  cents  a 
week,           ___-_-- 

* 

1 

ct. 
50 

Dr. 


Anthony  Billings. 


1822. 
March 

April 
May 

4 
10 
6 
12 
12 
25 

To  1  barrel  of  cider,  and  barrel,        - 
Cash  paid  your  order  in  favour  of  G.  Gilbert, 
Sundries,     ------- 
ditto,        ------- 

i 

26 
68 
43 
54 

ct. 
75 
32 
04 
00 
60 
00 

ditto, 
My  order  on  Thomas  Gros.venor, 

Dr. 


Thomas  Grosvenor. 


1822. 

1 

ct. 

Jan'y 

15 

To  the  frame  of  a  house, 

10000 

25 

Sundries,     ------- 

74|30 

Feb'y 

16 

Sundries,     -         - 

7 

11 

28 

The  frame  of  a  barn,    ----- 

75 

00 

April 

22 

Sundries,     -         -         -         -'- 

2|24 

Dr. 


Edward  Jones. 


1822. 
Feb'y 

1 
24]To  my  draft  on  Thomas  Grosvenor, 

38 

ct. 
00 

Dr. 


Peter  Daboll. 


18**.  I 


nnaries, 
\ 


$  \ct. 
5l51 


FORM  OF  A  LEGEK. 


A  hired  lad, 


Cr. 


1822. 
Jan'y 
May 

l,i 

15 

By  3  months'  wages  due  this  day,  at  $6,    - 
4i  months'  wages,  at  $7,      - 

18 
31 

ct> 
00 
50 

Farmer,                                Cr. 

Merchant,                              Cr. 

1822. 
Jan'y 

Feb'v 
April 

5 

26 
28 
29 

By  my  order  in  favour  of  Joseph  Hastings, 
Sundries,     ------- 
*  ditto,        _--_..- 
ditto,        ------- 

11 
3 
3 
9 

50 
62 
55 
99 

Judge  of  County  Court,                   Cr. 

Feb'y 
March 

April 

May 

"« 

12 

25 

t^  f  Border  in  favour  of  Joseph  Hastings, 
E'flMlraft  in  favour  of  Edward  Jones,    - 
"   /fi  paid  me  this  day,         - 
1  empty  cider  barrel, 
Amount  of  your  order  on  Theodore  Barrel], 
My  order  in  favour  of  Anthony  Billings, 

$3 
38 
75 

68 
54 

50 
00 

00 
58 
00 
00 

Labourer,                             Cr. 

1822. 
Jan'y 
Feb'y 

18  By  team  hire  at  sundry  times, 
18  1     4  months'  hire  of  his  son  William,  at  $10,      - 

5 
40 

ct. 
64 
00 

Farmer,                               Cr. 

1822.  1 
March!  15 

By  sundries  in  full, 

5 

51 

Dr. 


FORM  OF  A  LEGER. 

Samuel  Green. 


1822. 
March  28|To  cash  in  full  of  his  account, 


$  let 
3|81 


Dr. 


Notes  Payable. 


1822. 
Sept. 


24J  To  cash  paid  for  my  note  to  D.  Mason,     - 


$    ct. 
4800 


Dr. 


Jonathan  Curtis. 


1822.  1 
March  28 

To  a  bay  horse,       - 
A  wagon  and  harness, 

1     |    ct. 
2300 
-     1     4200 

John  Rogers. 


25  To  1  yoke  of  oxen  at  60  days'  credit, 


Theodore  Barrell. 


,May|  3|To  16  cheese,  weight  308  Ibs.  at  5  cents,    - 

#15 

40 

1 

217  Ibs.  butter  at  15  2-3  cents,       - 

34 

00 

24  Ibs.  honey  at  12^  cents,                 „ 

3 

00 

52 

40 

INDEX  TO  THE  LEGER. 


B. 

Barrell,  Theodore,     - 
Billings,  Anthony, 

PAGE 

2 
1 

H. 

Hastings,  Joseph, 

PAGE 
1 

J. 
Jones,  Edward, 

1 

C. 

Curtis,  Jonathan, 

2 

N. 
Notes  Payable, 

2 

D. 

Daboll,  Peter,    - 

1 

R. 

Rogers,  John,    - 

2 

G. 

Grosvenor,  Thomas, 

Oreon.  Samuel. 

1 

- 

S. 

Stacy.  Samuel,                              I 

f  • 

3~23^ 


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